UC-NRLF 


DISCUSSION 


OF   THE 


PRECISION  OF  MEASUREMENTS. 

WITH  EXAMPLES  TAKEN  MAINL  Y  fROM 

PHYSICS  AND   ELECTRICAL  ENGINEERING. 


BY 


SILAS  W.  HOLMAN,  S.B., 

l.ATE  ASSOCIATE  PROFESSOR  OF  PHYSICS, 
MASSACHUSETTS    INSTITUTE    OF  TECHNOLOGY. 


SECOND   EDITION^   REVISED. 
FIRST   THOUSAND. 


NEW  YORK : 

JOHN  WILEY  &  SONS. 

LONDON:  CHAPMAN  &  HALL,  LIMITED. 

1904. 


COPYRIGHT,  1892, 

BY 

SILAS  W.  HOLMAJ*. 


ROBERT  DRUMMOND,    ELECTROTYPER   AND   PRINTER,   NEW  YORK. 


PREFACE. 


THE  material  presented  in  this  volume  is  the  outcome  of 
several  years'  teaching  of  the  subject.  In  a  less  complete 
form  it  was  prepared  for  lecture  notes  and  was  printed  in 
pamphlet  form,  but  not  published,  by  the  Massachusetts  In- 
stitute of  Technology  in  1888,  having  appeared  in  the  Tech- 
nology Quarterly  and  in  the  Electrical  Engineer  in  1887. 

In  this  revised  form,  the  author  has  felt  that  it  perhaps 
possessed  sufficient  completeness  and  originality  to  be  of  in- 
terest or  value  to  students  and  teachers,  and  therefore  to 
merit  publication. 

In  venturing  to  urge  the  importance  of  the  subject  as  a 
course  of  study  for  engineers  and  for  students  of  physics  or 
other  pure  sciences,  the  author  would  suggest  the  value  of  the 
attitude  of  mind  produced  by  it.  One  who  has  in  any  reason- 
able degree  mastered  its  methods,  although  he  may  never  apply 
them  directly,  will  not  only  have  increased  his  power  to 
intelligently  scrutinize  experimental  results,  but  will  have 
acquired  a  tendency  to  do  so.  And  it  is  perhaps  not  too  much 
to  hope  that  he  may  acquire  a  notion  of  a  judicious  distribution 
of  effort  which,  with  the  best  of  results  to  himself,  he  may  carry 
into  quite  other  matters. 

SILAS  W.  HOLMAN. 

MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY, 
BOSTON,  September,  1892. 


PREFACE   TO   THE    SECOND   EDITION. 


IN  the  present  edition  some  errors  have  been  corrected  and 
a  few  minor  changes  have  been  made. 

S.  W.  H. 

June,  1897. 


268945 


CONTENTS. 


PRECISION   OF   MEASUREMENTS. 

PAGE 

Introductory I 

DIRECT  MEASUREMENTS. 

Direct  Measurements 4 

Indirect  Measurements 4 

Quantities :  Independent,  Conditioned 5 

Sources  of  Error 5 

Errors  of  Single  Observations 6 

Variable  Part 6 

Constant  Part,  Constant  Error 7 

Elimination  of  Constant  Error 7 

Corrections 8 

Example  I.     A,B,C.     Distance  by  Steel  Tape 9 

Determinate  and  Indeterminate  Errors 10 

Residuals II 

Accuracy  or  Error  of  Result 13 

Deviations 14 

General  Law  of  Deviations f 15 

Mean:  Best  Representative  Value 16 

Deviation   Measure 16 

Average  Deviation , 16 

Example  II 18 

Places  of  Figures  in  d.m. ;    and  Negligible  Amounts , 20 

Best  Value  of  n 22 

Other  Deviation  Measures 23 

Special  Law  of  Deviations , 24 

Precision  Measure  of  Result 25 

To  Make  Residuals  Negligible  in  P. M , 26 

Criterion 26 

Best  Value  of  Residuals:  Equal  Effects 27 

Fractional  Deviation,  Fractional  Precision 29 

Mistakes 30 

v 


Vi  CONTENTS. 

PAGE 

Criterion  for  Rejection  of  Doubtful  Observations 30 

Weights .' 31 

Meaning  of  Estimated  Accuracy  of  Direct  Result 32 

Forms  of  Problems  on  Accuracy  of  Result 33 

Data  Required  to  Substantiate  Result 36 

Planning  of  Direct  Measurement 36 

Solutions  of  Illustrative  Problems  in  Direct  Measurements 37 

Example  III.     Weighing.     Balance 37 

Example  IV.     Voltmeter  Calibration 41 

INDIRECT  MEASUREMENTS. 

Estimate  of  Accuracy  of  Indirect  Result 45 

Error  of  Method 46 

Check  Methods 47 

Relation  between  P.M.  of  Results  and  of  Components 47 

Types  of  Problems 47 

General  Formulae 48 

Notation 49 

Separate  Effects.     I,  II.     Formulae 49 

Resultant  Effects.     Ill;  1,2.     Formulas 50 

Equal  Effects.     Formulae 53 

Application  to  Precision  Discussions 54 

Formulas  for  General  and  Special  Functions 55 

Simple  Functions 56 

Separation  into  Factors  which  are  Functions  of  Single  Components. . .  61 

Separation  into  Groups 63 

Criteria  for  Negligibility  of  d  in  Components 67 

Numerical  Constants 70 

Equal  Effects.     Demonstration 70 

Estimated  Precision  Measures  of  Components 72 

Components  with  Special  Laws  of  Deviations 73 

Preparation  of  Functions  for  Discussion 73 

Simplification  of  Functions 75 

Significant  Figures 76 

Rules  for  Significant  Figures.   1-6 77 

Examples  V— XII 78 

Demonstration  of  Rules  80 

Forms  of  Problems  on  Accuracy  of  Result 84 

Data  Required  to  Substantiate  Result  85 

Planning  of  Indirect  Measurement 85 

Examples: — 

XIII— XVI.     Value  of  g  by  Simple  Pendulum 86 

XVII.     Calorimeter 88 

XVIII.     Heat  by  Incandescent  Lamp 89 


CONTENTS. 


Vll 


XIX.  Volume  of  Sphere 90 

XX.  Value  of  g  by  Simple  Pendulum 90 

XXI.  Cosine  Galvanometer 91 

XXII.  Continuous  Calorimeter 94 

XXIII.  H.  P.  by  Friction  Brake 96 

XXIV.  Specific  Resistance   ; 98 

BEST  MAGNITUDES  OF  COMPONENTS. 

Nature  of  Problems 100 

For  a  Single  Component , 102 

For  Two  Variable  Components 104 

Best  Ratio.      Procedure 104 

Best  Magnitudes 106 

For  Several  Components. ...    107 

Best  Ratio , 107 

Best  Magnitudes 108 

Approximate  Solution  by  Equal  Effects 108 

Best  Ratio 108 

Best  Magnitudes 109 

Examples: — 

XXV.     Best  Deflection  on  Tangent  Galvanometer no 

XXVI.     Electrical  Heating  of  Conductor in 

XXVII.     Bar  for  Moment  of  Inertia 112 

XXVIII.     Modulus  of  Elasticity  of  Wooden  Beam 115 

XXIX.     Specific  Resistance  of  Wire 118 

XXX.     XXVIII  by  Another  Method 118 

SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

Example  XXXI.  Calibration  of  Voltmeters 120 

Example  XXXU.  Dynamo  Efficiency  by  Stray-Power  Method 122 

Example  XXXIII.  Cradle  Dynamometer 130 

Example  XXXIV.  Tangent  Galvanometer, 138 

Example  XXXV.  Electro-static  Capacity.  Thomson's  or  GottY  Method  159 

Example  XXXVI.  Magnetometer 160 

Example  XXXVII.  Battery  Resistance  and  E.  M.  F -  161 

TABLES. 

Sines,  Cosines,  Tangents 166 

Constants 166 

Squares,  Cubes,  Reciprocals 167 

Logarithms   168 

INDEX 171 


PRECISION    OF    MEASUREMENTS. 


INTRODUCTORY. 

AN  experimental  result  whose  reliability  is  unknown  is 
nearly  worthless.  The  grade  of  accuracy  of  a  measurement 
must  be  adapted  to  the  purpose  for  which  the  result  is  desired. 
The  necessary  accuracy  must  be  secured  with  the  least  possible 
expenditure  of  labor. 

These  statements  apply  no  less  to  the  roughest  than  to  the 
most  elaborate  work  which  the  engineer  is  called  upon  to  per- 
form ;  they  are  no  more  true  of  refined  scientific  research  than 
of  every-day  engineering  and  industrial  practice.  The  prin- 
ciples which  underlie  these  assertions  respecting  quantitative 
measurement  differ  in  no  essential  particular  from  those  which 
lie  at  the  foundation  of  all  commercial  and  industrial  economy, 
— proved  value ;  product,  labor,  and  expenditure  proportioned 
to  the  relative  importance  of  the  thing  in  hand  ;  results  ob- 
tained with  the  least  effort,  and  hence  with  judicious  distribu- 
tion of  effort  among  the  various  parts  of  the  work. 

The  successful  business  manager  does  not  hesitate  at  large 
expenditure  of  money  or  effort  in  those  parts  of  an  under- 
taking where  he  perceives  them  to  be  necessary,  nor  does  he 
overlook  the  importance  of  economy  where  expenditure  is  not 
essential.  Neither  does  he  wait  till  an  enterprise  is  well  under 
way  or  completed  to  determine  where  the  chief  points  for  ex- 
penditure or  economy  lie.  The  wise  designer  of  a  structure 


2  INTRODUCTORY. 

devotes  his  close  attention  to  distributing  material  to  the  best 
advantage :  enough,  at  the  best  points,  and  no  superfluity. 
These  things  are  so  obvious,  and  their  neglect  is  so  strikingly 
absurd,  that  it  is  the  more  surprising  that  the  same  practice 
should  be  so  commonly  neglected  not  only  in  quantitative 
measurement  but  in  engineering  investigations  and  even  in 
physical  research. 

The  engineer,  in  the  measurement  of  the  efficiency  or  duty 
of  an  engine,  the  efficiency  of  a  dynamo  or  of  a  power  station; 
the  physicist  in  the  designing  or  use  of  a  gas  thermometer,  in 
the  measurement  of  an  index  of  refraction,  or  in  the  compari- 
sons of  standards  of  length  ;  the  chemist  in  analytical  investi- 
gation, or  in  the  experimental  test  of  an  industrial  plant, — 
can  no  more  afford  to  omit  a  preliminary  discussion  of  the 
precision  of  the  various  component  measurements  entering 
into  his  result,  than  the  business  man  to  omit  to  estimate  and 
proportion  in  advance  his  expenditure  in  a  large  undertaking. 
The  one  is  as  essential  as  the  other  to  complete  success. 

The  thoughtful  student  recognizes  early  in  his  experimental 
work  the  importance  of  certain  questions  which  never  leave  the 
mind  of  the  experienced  observer,  namely — What  accuracy  is 
desired  in  the  result  ?  What  accuracy  is  therefore  necessary 
in  each  of  the  various  component  measurements  from  which 
the  result  is  calculated  ?  How  reliable  is  the  final  result  when 
obtained?  The  more  complicated  and  indirect  the  measure- 
ment, the  more  difficult  it  becomes  to  answer  these  queries  by 
mere  inspection,  and  hence  the  greater  the  necessity  for  some 
systematic  and  rational  procedure  for  reaching  the  answer. 
The  present  volume  is  the  outcome  of  an  effort  to  establish 
such  a  procedure  which,  while  being  sufficiently  general,  shall 
not  be  too  laborious  in  its  operation.  It  is  intended  to  be 
applicable  to  quantitative  measurements  of  all  kinds,  whether 
in  engineering  or  pure  science.  The  illustrative  examples 
throughout  the  text  are  taken  chiefly  from  physics  and  elec- 
trical engineering,  because  the  students  as  well  as  the  problems 
with  which  the  author  has  been  called  upon  to  deal  have  been 
chiefly  in  those  subjects.  The  examples  are  for  the  most  part 


IN  TROD  UC  TOR  Y.  3 

so  fully  explained  or  so  simple  that  they  will  be  easily  intelli- 
gible to  students  in  other  lines.  The  processes  of  the  differen- 
tial calculus  have  been  used,  because  without  them  the  methods 
would  necessarily  be  cumbrous,  and  also  because  a  large  and 
increasing  proportion  of  those  who  deal  at  all  with  such  a* 
subject  are  amply  competent  to  follow  or  make  such  simple 
differentiations  as  are  required.  It  is,  however,  to  be  noted 
that  the  majority  of  the  methods  and  formulae  herein  de- 
veloped can  be  utilized  without  any  employment  whatever 
of  the  calculus,  so  that  they  may  be  applied  bj'  one  who  has 
forgotten  his  earlier  knowledge  of  that  subject  or  who  has 
never  become  acquainted  with  it.  Attention  is  particularly 
directed  in  this  connection  to  the  rules  for  significant  figures. 


DIRECT    MEASUREMENTS. 


Direct  Measurements. — All  quantitative  work  of  course 
involves  measurements.  These  may  be  separated  into  two 
classes,  viz.  direct  and  indirect.  Direct  measurements  are 
those  made  by  methods  and  instruments  whose  indications 
give  directly  the  quantity  sought ;  e.g.  measurements  of  dis- 
tance by  a  scale,  of  weights  (or  masses)  by  an  equal-arm  bal- 
ance, of  resistance  by  a  Wheatstone  bridge,  etc.  The  direct 
readings,  in  such  cases,  may  or  may  not  require  corrections. 
The  fact  that  a  correction  is  necessary,  that  is,  that  the  directly 
observed  value  must  be  more  or  less  modified  to  remove  the 
effect  of  known  sources  of  error,  does  not  render  the  measure- 
ment indirect. 

Indirect  Measurements  are  those  in  which  the  quantity 
measured  is  not  given  directly  by  observation  or  readings 
taken,  but  must  be  calculated  from  them.  Thus  in  an  indirect 
measurement  the  quantity  sought  is  a  function  of  one  or  more 
quantities  which  are  directly  measured  and  which  may  be 
called  the  component  quantities.  For  instance,  the  specific 
gravity  of  a  substance  is  ordinarily  found  by  measuring  its 
weight  in  air  and  its  loss  of  weight  in  water.  Each  of  these  is 
or  may  be  a  direct  measurement,  but  the  desired  specific 
gravity  is  found  from  them  by  calculation,  viz.  by  dividing 
one  by  the  other,  and  is  therefore  indirect.  To  measure  a 

4 


SOURCES  OF  ERROR  IN  DIRECT  MEASUREMENTS.         5 

constant  current  C,  we  may  pass  it  through  a  tangent  galva- 
nometer of  factor  k  and  observe  the  deflection  0  produced : 
then  C=  k  tan  0.  Here  C  is  indirectly  measured,  being  cal- 
culated from  the  directly  observed  deflection  0  by  the  func- 
tion indicated  by  the  right-hand  member  9f  the  expression. 
Similarly  the  measurement  of  g  by  a  simple  pendulum,  of  the 
E.  M.  F.  of  a  battery  by  the  two-deflection  method  or  by  the 
Poggendorff  method,  of  the  index  of  refraction  of  a  prism, 
of  the  efficiency  of  a  dynamo  or  motor, — in  fact  the  great 
majority  of  physical  measurements, — are  indirect. 

Quantities  may  be  either  independent  or  conditioned* 
That  is,  two  or  any  number  of  quantities  to  be  measured  may 
be  wholly  independent,  so  that  the  magnitude  of  one  is  in  no 
way  predetermined  by  any  relations  to  the  others ;  or  they 
may  be  conditioned  so  that,  for  instance,  the  magnitude  ol 
two  out  of  three  being  given,  that  of  the  third  is  thereby  pre 
determined.  Thus  a  constant  current  flowing  through  a  given 
coil  of  wire  might  be  anything  whatever,  according  to  the 
potential  used,  so  that  measurements  of  current  and  of  resist- 
ance at  any  instant  would  be  independent.  But  if  the  poten- 
tial difference  at  the  ends  of  the  coil  were  measured  simulta- 
neously with  the  current  and  resistance,  then  the  three,  current, 
resistance,  and  potential,  would  be  conditioned  by  Ohm's  law. 
The  numerical  values  obtained  in  the  measurement  of  condi- 
tioned quantities  contain,  of  course,  errors  not  controlled  by 
these  conditions,  so  that  these  values  fail  to  fulfil  the  condi- 
tions, and  require  adjustment. 

Sources  of  Error  in  Direct  Measurements. — All  processes 
of  measurement  are,  of  course,  fallible.  None  can  give  abso- 
lute accuracy,  that  is,  none  can  be  wholly  free  from  error.  The 
questions  with  which  we  have  to  deal  then  are  only  such  as 
relate  to  the  amount  or  character  of  the  errors  occurring,  and 
to  their  sufficient  elimination  for  the  purpose  in  hand. 

Inspection  of  the  methods,  instruments,  and  results  of  any 
direct  measurement  will  show  that  the  method  has  some  dis- 
coverable sources  of  error,  that  the  instruments  likewise  contain 
certain  inherent  sources  of  error,  and  finally  that  however  care- 


6  DIRECT  MEASUREMENTS. 

fully  the  effects  of  the  discoverable  sources  are  removed,  some 
undiscovered  or  uncorrected  sources  still  remain,  since  succes- 
sive equally  careful  repetitions  of  the  same  measurement  yield 
numerical  results  which  are  more  or  less  discordant  in  the  last 
one  or  two  places  of  significant  figures. 

Example  I  (A),  page  9. 

The  existence  of  this  discordance  just  referred  to  proves 
that  the  errors  from  the  various  sources  are  not  constant,  at 
least  that  some  of  them  are  not, — a  fact  which  we  know  to  be 
true  for  some  of  the  discoverable  sources.  And  the  general 
rule  for  the  variation  doubtless  is  that  under  given  conditions 
the  error  from  any  given  source  has  a  certain  average  magni- 
tude about  which  it  varies  more  or  less,  being  sometimes 
greater  sometimes  smaller  than  that  amount.  It  is  therefore 
reasonable,  and  will  be  found  convenient,  to  regard  the  error 
from  any  source  as  made  up  of  two  portions,  a  constant  part, 
viz.  its  average  value,  and  a  variable  part.  Of  course  either  of 
these  parts  may  be  wanting  in  any  given  instance. 

Errors  of  Single  Observations. — The  actual  error  of  any 
given  single  observation  is  obviously  the  algebraic  sum  of  all 
the  individual  errors  from  the  several  sources  which  affect  the 
quantity.  As  these  individual  errors  have  each  a  constant  part 
and  a  variable  part,  so  the  error  of  a  single  observation  will  be 
made  up  of  two  parts.  These  will  be  a  constant  portion  which 
is  the  algebraic  sum  of  the  constant  parts  of  the  individual 
errors,  and  a  second  portion  which  will  vary  in  different  obser- 
vations and  which  is,  for  any  observation,  the  algebraic  sum 
of  the  variable  parts  of  the  individual  errors  as  they  existed 
at  the  moment  when  that  observation  was  made. 

Variable  Part. — Considering  first  this  part  of  the  error,  we 
can  at  once  see  that,  if  we  make  a  series  of  observations  under 
sensibly  the  same  conditions  and  take  the  average,  the  result 
will  be  partly  free  from  the  effect  of  the  variable  parts  of  the 
error.  For  each  varying  error  will  tend  to  make  the  result  at 
one  time  more  or  less  too  large,  at  another  too  small, — a  kind 
of  fluctuation  which  the  process  of  averaging  tends  to  elimi- 
nate. That  the  arithmetical  mean  removes  the  variable  parts 


ELIMINA  TION  OF  CONSTANT  ERROR.  / 

of  the  error  better  than  any  other  function,  will  be  shown 
more  explicitly  later.  It  is  easy  to  see  that  averaging  can 
never  effect  a  complete  removal ;  the  elimination  being,  how- 
ever, more  nearly  complete  (in  proportion  to  Vn)  as  the  num- 
ber, n,  of  observations  is  greater.  For  the  sum  total  of  the 
positive  parts  of  the  variation  will  naturally  be  unequal  to 
that  of  the  negative  parts,  and  so  long  as  the  number  is  small 
the  inequality  will  be  considerable,  but  they  will  become  more 
and  more  nearly  equal  as  the  number  increases.  At  best, 
however,  there  will  always  remain  a  residual  variable  error,  and 
in  discussing  the  correctness  of  the  result  this  must  be  taken 
into  account. 

Example  I  (B\  page  9. 

Constant  Part. — Considering  next  the  effect  of  the  con- 
stant parts  of  the  errors  from  the  various  sources,  we  see  that 
their  resultant,  viz.,  the  algebraic  sum  of  all  these  constant 
parts,  will  itself  necessarily  be  of  the  same  amount  in  each  and 
all  the  si'ngle  observations  taken  under  the  same  conditions. 
The  process  of  averaging  a  series  will  therefore  do  absolutely 
nothing  toward  the  removal  of  this  resultant  of  the  constant 
portions.  The  mean  will  contain  an  error  of  which  the  con- 
stant portion  will  be  identical  with  that  of  each  single  observa- 
tion in  the  series.  The  name  "  constant  error "  is  therefore 
well  applied  to  this  resultant  constant  portion  of  the  errors. 

Elimination  of  Constant  Error. — Of  the  constant  portions 
of  the  individual  errors  going  to  make  up  this  resultant  con- 
stant error,  some  will  be  positive,  some  negative,  and  the 
magnitudes  will  be  various.  They  will  therefore  in  part  annul 
one  another;  that  is, the  resultant  will  not  in  general  be  as  large 
as  the  arithmetical  sum  of  all  the  component  parts.  If  the 
sum  of  the  positive  parts  exactly  equalled  that  of  the  negative 
parts,  the  elimination  would,  of  course,  be  complete,  and 
the  constant  error  of  the  single  observation  and  mean  would 
be  zero.  But  this  condition  would  naturally  be  highly 
exceptional.  The  constant  error  is  in  fact  exceedingly 
difficult  of  removal,  and  often  proves  to  be  of  surprisingly 
large  amount  in  spite  of  most  painstaking  efforts  for  its  elim- 


8  DIRECT  MEASUREMENTS. 

ination.  For  any  specific  method  or  apparatus  will  have  its 
own  characteristic  set  of  errors,  of  which  some  will  be  pre- 
dominant and  will  determine  a  constant  error  more  or  less 
large  in  the  results  obtained.  Another  method  will  on  the 
other  hand  be  characterized  by  a  different  set  of  sources  of 
error  and  will  have  a  different  constant  error.  Observations 
taken  under  diverse  conditions  with  the  same  method  will 
often  also  have  differing  constant  errors.  And  finally,  differ- 
ent observers  will  show  different  "  personal  equations."  Thus, 
results  obtained  by  changing  methods,  apparatus,  observers, 
or  other  conditions  will  materially  differ  in  the  sources  and 
amounts  of  their  constant  errors.  The  greater  the  number 
and  the  more  complete  the  variety  of  the  changes,  the  greater 
becomes  the  diversity  of  the  sources  of  error,  and  conse- 
quently the  more  complete  the  elimination  of  their  effects  by 
taking  a  general  mean  (with  due  respect  to  weights)  of  the 
various  results.  It  is  only  through  this  repetition  by  inde- 
pendent methods  that  we  can  gain  confidence  as  to  the  real 
accuracy  of  a  result ;  and  the  more  radically  distinct  the 
nature  of  the  methods  employed,  the  more  valuable  the  check. 
Even  a  single  reliable  check  greatly  enhances  the  value  of  a 
result.  It  is  evident  that  we  can  obtain  no  numerical  meas- 
ure of  the  amount  of  the  constant  error  of  any  result.  Such  a 
measure  would  imply  the  knowledge  of  the  true  result,  which 
is  of  course  always  unknown. 

Example  I  (C\  page  10. 

Correctives. — In  as  much  as  each  method  and  apparatus 
has  its  own  characteristic  sources  which  determine  its  constant 
error,  it  is  obvious  that  in  using  any  method  we  must  do  what 
we  can  to  reduce  the  effect  of  those  sources  to  a  minimum. 
For  this  purpose  we  must  study  the  method  and  instruments 
as  thoroughly  as  possible  in  advance,  to  discover  all  possible 
sources  of  error.  We  must  then  arrange  the  work  to  remove 
as  many  as  possible  of  these  sources  wholly  or  in  part,  and  we 
must  evaluate  the  effects  of  those  not  removed  and  eliminate 
them  by  the  application  of  the  corresponding  "corrections  " 
thus  determined. 


EXAMPLE  I.  9 

Example  J.  (See  pages  6,  7  ^  seg.) — Take  as  an  illustra- 
tion so  simple  a  measurement  as  that  of  the  distance  between 
two  points  by  means  of  a  steel  tape. 

(A)  There  are  easily  discoverable  such  sources  of  error  as 

these  : 

(1)  Error  in  numbering  of  tape  ; 

(2)  Irregular  spacing  of  divisions  ; 

(3)  Incorrect  unit,  i.e.,  foot  not  standard  in  length: 

(4)  Bends  in  tape  ; 

(5)  Sag  of  tape  ; 

(6)  Stretch  of  tape  • 

(7)  Error  in  setting  zero  of  tape  at  starting  point ; 

(8)  Error  of  estimation  of  fraction  of  division  at  finishing 
point  ; 

(9)  Temperature  not   that  for  which  the  tape  was  gradu- 
ated. 

Besides  these  sources  there  are  doubtless  many  others  of 
greater  or  less  effect,  some  of  which  might  possibly  be  dis- 
covered by  further  study,  but  many  of  which  are  at  present 
obscure. 

Successive  measurements  of  the  same  distance,  especially 
if  this  be  long  and  if  the  fraction  of  an  inch  to  which  readings 
are  taken  be  small,  will  show  discordances  of  greater  or  less 
magnitude. 

(B)  Variable  Part.      Errors  5,  6,   7,    8,  9  would  vary   in 
amount  from   time   to  time  and  between  different  readings, 
and  would  therefore  have  variable  parts.     Each  would  tend 
to  make  the  single  results   sometimes  larger,  at  other  times 
smaller,  and  by  irregular  amounts.     Thus  in  the  average  re- 
sult of  a  series  of  observations  the  variable  parts  of  the  error 
from  any  single  source  would  in  part  annul  itself.     Also  in  any 
single  observation  the  sum  of  the  negative  variable  parts  of 
the  errors  from  all  sources  would  offset  in  part  the  sum  of  the 
positive  variable  parts   more  or   less  completely,  but   seldom 
wholly. 

(C)  Constant  Error.     The  errors   i,  2,   3,  and  4  would  be 
constant  for  any  given  distance ;    also,   5   and  6  will    clearly 


10  DIRECT  MEASUREMENTS. 

be  liable  to  have  some  constant  portion,  as  also  would  9 
under  some  circumstances.  These  together  will  make  up  the 
constant  error.  Some  will  be  of  one  sign,  some  of  the  other, 
so  that  they  will  in  part  neutralize,  but  cannot  be  expected  to 
wholly  do  so.  The  separate  constant  portions  are  the  same  in 
all  single  observations.  Hence  the  constant  error  will  be  the 
same  in  each  single  reading  and  in  the  mean  result. 

The  part  of  the  constant  error  due  to  5  and  6  may  be 
largely  removed  by  stretching  the  tape  by  a  spring-balance  or 
other  means  so  that  it  is  always  under  the  same  tension.  A 
correction  for  the  amount  of -sag  can  then  be  made,  and  the 
stretch  will  be  nearly  the  same  at  all  times.  The  error  from  9 
may  be  in  part  removed  by  measuring  the  temperature  of  the 
tape  at  various  points  along  its  length  and  correcting  for  the  ex- 
pansion due  to  the  difference  between  the  observed  tempera- 
ture and  that  at  which  the  tape  is  correct.  There  is  liability  in 
this  correction  to  a  residual  constant  error  due  to  uncertainty 
as  to  the  value  of  the  coefficient  of  expansion  for  the  metal 
employed,  and  due  to  constant  errors  in  the  thermometers. 
There  is  liability  to  variable  error  from  the  thermometers 
being  too  far  apart  or  improperly  located  to  give  the  true 
mean  temperature  of  the  tape  ;  also  from  the  variable  errors 
of  the  thermometers  themselves. 

It  is  clear,  upon  reflection,  that  the  errors  \\\  the  above 
measurement,  especially  the  constant  errors,  are  largely 
peculiar  to  the  method.  If  the  distance  were  to  be  measured 
by  a  rod,  or  by  a  base-line  apparatus,  or  by  stadia  wires  in  a 
telescope,  each  of  the  results  would  be  characterized  by  a 
different  set  of  errors,  of  which  some  might  be  common  to  all. 
The  mean  of  the  results  of  such  different  methods  would  be 
certainly  more  reliable  than  the  poorest  of  them,  by  the 
natural  annulling  of  the  different  classes  of  errors. 

Determinate  and  Indeterminate  Errors. — All  sources 
of  error  which  are  discoverable  and  which  may  be  removed  or 
may  have  their  effects  more  or  less  completely  allowed  for  by 
corrections  will  here  be  classed  as  determinate  sources.  The 
corresponding  errors  will  be  referred  to  as  determinate  errors. 


RESIDUALS.  U 

On  the  other  hand,  all  sources  which  are  either  undiscover- 
able,  or  whose  effects  cannot  be  properly  determined  and 
allowed  for,  will  be  classed  as  indeterminate  sources,  and  the 
corresponding  errors,  as  indeterminate  errors.  This  class  will 
contain  not  only  those  which  are  undiscoverable,  but  also  the 
residuals  of  determinate  errors.  Both  determinate  and  inde- 
terminate sources  are  inevitably  present  in  every  direct,  and 
therefore  also  in  every  indirect,  measurement. 

Residuals. — In  general  the  processes  for  the  elimination 
of  the  determinate  errors,  whether  by  the  removal  of  their 
sources  or  by  corrections,  accomplish  this  object  only  approxi- 
mately. There  are  perhaps  a  few  cases  in  which  the  source  of 
a  determinate  error  can  be  wholly  removed,  or  at  least  to  a 
far  greater  extent  than  is  demanded.  For  instance,  in  the 
constant  n  we  may  retain  so  many  places  of  figures  that  the 
error  from  rejecting  the  rest  may  be  utterly  insignificant 
Ordinarily,  however,  the  source  cannot  be  removed,  but  its 
effect  can  be  lessened  so  as  to  be  small  or  negligible.  For 
instance,  the  individual  weights  of  a  set  can  perhaps  be 
adjusted  so  accurately  that  their  errors  are  negligible  for  a 
purpose  in  hand ;  or  the  arms  of  a  balance  may  be  made  so 
nearly  equal  that  the  error  is  negligible.  But  in  all  such  cases 
there  remains  an  error  more  or  less  small  which  enters  into 
the  result  and  which  will  be  called  a  residual. 

A  residual  may  be  insignificant,  but  this  requires  proof ;  and 
the  proof  can  only  be  arrived  at,  in  general,  by  a  direct  meas- 
urement of  some  kind,  such  as  a  comparison  with  a  standard, 
or  a  measurement  of  some  ratio.  For  instance,  the  weights  of 
the  set  can  be  assumed  to  have  a  negligible  error  only  after 
each  has  been  weighed  against  a  standard  or  tested  by  some 
equivalent  process.  This  weighing  will  be  made  only  to  a  cer- 
tain grade  of  accuracy,  and  will  therefore  itself  leave  a  residual 
error.  Similarly  the  ratio  of  the  arms  of  the  balance  must  be 
determined  by  the  usual  process  of  balancing  and  interchang 
ing  equal  masses.  Therefore  this  also  is  a  direct  measurement, 


12  DIRECT  MEASUREMENTS. 

and  will  of  course  be  made  only  to  the  limit  of  accuracy 
fixed  by  the  sensitiveness  of  the  balance,  and  will  leave  a 
residual  error. 

The  process  of  evaluation  of  a  correction  is  in  general 
also  only  an  approximate  one,  and  consists  usually  of  a  direct 
or  indirect  measurement  carried  out  only  with  a  certain 
degree  of  accuracy  leaving  a  residual  error.  Thus  in  the 
foregoing  examples  the  weights  might  be  adjusted  less  closely 
than  was  demanded  for  the  work  in  hand,  and  the  corrections 
to  be  applied  might  be  evaluated  by  weighing  against  standard 
weights  and  thus  determining  the  error.  But  this  weighing 
would  be  a  direct  measurement,  and  would  be  carried  out  to 
an  accuracy  limited  by  the  sensitiveness  of  the  balance  or  by 
some  other  conditions.  A  corresponding  residual  error  would 
therefore  be  left  after  the  correction  was  applied.  Similarly, 
the  ratio  of  the  balance  arms  not  being  close  enough,  it  might 
be  allowed  for  by  measuring  the  ratio  and  applying  a  correc- 
tion. This  again  would  leave  a  residual  error. 

Other  processes  of  correction  exist,  such  as  the  correction 
for  the  eccentricity  of  a  circle  by  reading  two  verniers  180° 
apart,  and  averaging.  This  is  a  type  of  certain  mathematical 
corrections,  and  these  also  are  usually  only  approximate,  being 
close  enough  when  the  errors  are  small,  but  nevertheless  leav- 
ing residual  errors. 

In  brief,  then,  most  processes  for  the  elimination  of  deter- 
minate errors  leave  residual  errors  behind.  Also,  most  such 
processes  involve  direct  measurements,  and  the  statements 
already  made  or  to  be  made  respecting  direct  measurements 
apply  to  them.  The  numerical  measure  of  the  residuals  will 
in  general  therefore  be  of  the  nature  of  precision  measures  of 
direct  measurements  which  will  presently  be  discussed. 

From  this  it  is  obvious  that  if  it  were  necessary  in  an 
investigation  to  work  out  from  the  beginning  every  detail  of  a 
research,  establishing  all  standards,  ascertaining  all  correc- 
tions, developing  every  process  employed,  the  labor  would  be 
enormous — as,  indeed,  it  often  is.  But  fortunately  the  prog- 
ress of  experimental  science  has  provided  instruments,  pro- 


ACCURACY  OR  ERROR  OF  RESULTS.  13 

cesses,  methods,  and  results  of  known  accuracy,  which  may  be 
appropriated  in  any  desired  manner  in  more  complex  investi- 
gations. 

Accuracy  or  Error  of  Results. — By  the  accuracy  of  a  re- 
sult we  mean  its  freedom  from  error.  The  real  measure  of 
the  accuracy  of  a  result  is  therefore  the  error  of  that  result. 
Thus  if  we  knew  that  a  result  had  an  error  of  2  per  cent  we 
should  say  that  it  was  accurate  to  2  per  cent,  or  we  might 
say  that  its  accuracy  was  98  per  cent.  The  latter  phrase, 
although  more  exact,  is  less  common  and  convenient  than  to 
say  that  the  accuracy  was  2  per  cent.  Thus  if  a  result  were 
24.967  metres,  and  were  known  to  have  an  error  of  0.025 
metres,  we  should  say  the  result  was  accurate  to  0.025  m.  or 
to  o.i  per  cent;  or  we  might  say  that  it  had  an  accuracy  0/ 
O.I  per  cent,  although  that  phrase  would  be  less  precise  than 
to  say  that  it  had  an  accuracy  of  99.9  per  cent. 

But  it  is  clear  that  we  can  have  no  numerical  measure  of 
the  constant  error  of  a  result,  whether  that  result  be  a  single 
observation,  a  mean  of  a  series  by  one  method,  or  the  mean  of 
results  by  a  large  number  of  methods.  For  as  the  error  is  the 
amount  that  the  measured  result  differs  from  the  true  value, 
such  a  measure  necessarily  implies  that  the  true  value  of  the 
quantity  is  known,  which  is  never  the  case. 

Yet  it  is  of  the  utmost  importance  that  we  should  be  able 
to  form  some  estimate  of  the  accuracy  or  of  the  error  of  the 
result,  and  that  this  should  be  expressed  numerically,  so  far  as 
possible.  How  such  an  estimate  is  arrived  at  will  be  here 
indicated,  and  just  what  the  measure  is  will  be  more  explicitly 
stated  in  a  later  paragraph. 

There  are  only  two  things  upon  which  this  estimate  can  be 
based,  in  the  case  of  the  result  of  a  series  of  observations  by  a 
single  method,  viz. : 

(1)  The  degree  of  care  exercised  in  the  study  and  removal 
of  the  determinate  errors. 

(2)  The  concordance,  or  rather  the  discordance,  between 
the  single  observations  of  the  series. 

Of  the  first,  we  have  a  partial  numerical  measure  in   the 


14  DIRECT  MEASUREMENTS. 

measure  of  the  residuals  of  the  determinate  errors,  but  this  is 
only  partial.  A  judgment  as  to  the  sources  of  error  which 
have  been  overlooked  or  neglected  is  essential,  but  this  cannot 
be  given  a  numerical  expression. 

Of  the  second,  the  numerical  measure  is  the  "  deviation 
measure  "  to  be  presently  described. 

The  deviation  measure  and  residuals  can  be  combined  to 
give  the  **  precision  measure  "  which  is  the  final  numerical 
measure  to  which  we  are  brought  in  forming  our  estimate,  and 
of  which  the  significance  will  be  stated  later. 

Thus  the  most  that  can  be  done  in  forming  an  estimate  of 
the  error  of  the  result  of  the  mean  of  a  series  of  observations 
by  a  single  method  is  this  :  the  precision  measure  of  the 
result  is  calculated,  giving  a  partial  numerical  measure  ;  and 
a  judgment  is  formed  from  an  inspection  of  the  method  as  to 
whether  any  constant  error  comparable  with  the  precision 
measure  probably  exists  in  the  result. 

If  results  by  several  different  methods,  etc.,  are  available, 
the  best  representative  value  (weighted  mean)  can  be  obtained 
from  them,  and  their  concordance  will  give  us  a  further  partial 
indication  of  the  correctness  of  that  value. 

It  becomes  necessary,  therefore,  to  discuss  the  meaning  of 
the  terms,  and  to  fix  upon  certain  points  respecting  deviations 
and  their  measure,  and  the  precision  measure. 

Deviations.  —  Suppose  any  number,  »,  of  direct  measure- 
ments or  observations  of  a  quantity  to  have  been  made  with 
equal  care,  and  under  apparently  identical  conditions.  Let 
alt  a,  ,  .  .  .  ,  an  represent  the  separate  results.  Let  A  represent 
their  arithmetical  mean  or  average.  Then  the  differences  of 
these  from  the  mean  will  be  given  by 


These  differences  will  be  called  the  deviations  of  the  single  ob- 
servations from  the  mean.  They  are  the  effects  of  the  vari- 
able parts  of  the  errors  affecting  the  measurements.  They  are 
not  the  errors  of  al  ,  #„  ,  etc.  ;  for  errors  are  the  discrepancies 


GENERAL  LAW  OF  DEVIATIONS.  15 

between  observed  and  true  values.  But  in  this  as  in  all  cases 
the  true  value  is  unknown,  and  the  deviations  are  merely  the 
differences  from  the  mean  value  A,  which  is  selected  as  being 
the  best  representative  value,  but  may  differ  much  from  the 
true  value.  Thus  the  deviations  measure  only  the  variable 
part  of  the  errors  and  give  no  clue  whatever  to  the  constant 
parts. 

General  Law  of  Deviations. — If  the  number,  #,  of  obser- 
vations m  the  series  be  very  great  (to  eliminate  exceptional 
irregularities),  it  is  found  as  the  result  of  the  study  of  actual 
series  of  observations  that  the  deviations  follow  a  definite  law, 
both  as  to  sign  and  magnitude.  This  law  is  apparently  the 
same  for  all  kinds  of  measurements  which  are  affected  by  a 
large  number  of  sources  of  error,  and  may  be  called  the  gen- 
eral law  of  deviations.  Special  laws  arise  in  certain  cases,  as 
will  be  further  indicated.  The  general  law  may  be  approxi- 
mately stated  in  words  thus :  Positive  and  negative  deviations 
of  any  given  magnitude  occur  with  equal  frequency ;  small  de- 
viations are  more  frequent  than  large  ones ;  very  large  devia- 
tions occur  very  seldom.  The  law  is  more  exactly  expressed 
by  the  equation 

y  =  ke-h**\ 

where  y  =  frequency  of  occurrence  of  deviation  whose  magni- 
tude has  any  assigned  value  x,  and  where  k  and  h  are  constants, 
and  e  is  the  base  of  the  Naperian  system  of  logarithms. 

This  expression  was  deduced  by  Laplace  by  an  a  priori 
mathematical  process  as  showing  the  probability  of  occurrence 
of  an  error  of  any  given  magnitude  when  the  error  was  not  of 
simple  origin,  but  was  produced  by  the  algebraic  combination 
of  a  great  many  independent  causes  of  error,  each  of  which, 
according  to  the  chance  which  affects  it  independently,  might 
produce  an  error  of  either  sign  and  of  different  magnitude. 
Applied  to  actual  series  of  observations  it  is  found  to  sensibly 
coincide  with  the  distribution  of  their  deviations.  This  expo- 
nential equation  may  then  be  held  as  representing  the  general 


l6  DIRECT  MEASUREMENTS. 

law  of  distribution  of  deviations,  being  in  accord  both  with  the 
theory  of  probabilities  and  the  results  of  experience.  It  is  sen- 
sibly exact  when  the  number  of  observations  is  large.  When 
the  number  is  small,  the  distribution  can  follow  this  law  only 
roughly,  but  no  other  law  would  be  more  closely  followed. 
The  approximation  with  which  the  series  of  observations  is 
represented  by  the  law  is  then  greater  the  larger  the  number 
of  observations  in  the  series. 

Mean:  Best  Representative  Value. — In  a  large  series  of 
equally  careful  observations  of  the  same  quantity,  under  the 
same  conditions,  the  variable  parts  of  the  errors  will  be  sensibly 
eliminated  by  averaging  the  results,  that  is,  by  the  employment 
of  the  mean  as  a  representative  value.  The  law  of  deviations 
already  stated  shows  that  to  be  true,  and  as  this  law  has  been 
arrived  at  by  an  application  of  the  theory  of  probabilities  and 
confirmed  by  the  results  of  specific  as  well  as  of  general  ex- 
perience, the  use  of  the  arithmetical  mean  as  the  best  repre- 
sentative value  in  such  a  large  series  can  be  considered  as  in 
accord  both  with  the  theory  of  probabilities  and  with  practical 
experience.  But  its  employment  is,  however,  justifiable  not  in 
in  large  series  only,  but  in  small  ones  as  well.  For  although 
the  reliability  or  degree  of  probability  of  the  mean  in  a  small 
series  will  be  less  than  in  a  larger  one,  yet  the  mean  has  a 
greater  probability  even  in  a  very  small  series  than  any  other 
representative  value  which  can  be  indicated. 

We  are  accustomed  to  think  of  the  mean  as  being  more 
reliable  in  proportion  to  the  square  root  of  the  number  of  ob- 
servations in  the  series,  but  we  must  avoid  attaching  undue 
weight  to  this  numerical  relation  when  the  number  of  observa- 
tions is  very  small,  as  for  instance  when  not  exceeding  five  or 
ten.  A  similar  caution  should  be  urged  respecting  all  applica- 
tions of  the  methods  and  rules  of  least  squares  when  n  is  small, 
although  the  use  of  the  methods  in  such  cases  is  fully  justi- 
fied by  the  fact  that  they  give  the  best  results  obtainable. 

Deviation  Measure,  Average  Deviation. — The  magnitudes 
of  the  deviations  in  a  given  series,  although  giving  no  indica- 
tion as  to  constant  errors,  do  furnish  a  measure  of  the  variable 


DE  VIA  TIQN  MEASURE,   A  VERA GE  DE  VIA  TION.  \ 7 

parts  of  the  errors,  since  it  is  to  these  that  they  are  due.  But 
where  the  number,  n,  of  observations  is  not  very  small,  mere 
inspection  does  not  readily  give  a  definite  idea  of  the  magni- 
tude of  the  deviations;  moreover  for  many  purposes  of  cal- 
culation it  is  necessary  to  have  a  single  number  to  represent 
them.  The  simplest  method  of  obtaining  such  a  number  is  to 
take  the  arithmetical  mean  of  the  deviations  without  respect 
to  sign,  that  is,  with  regard  to  magnitude  only.  This  quantity 
will  be  called  the  average  deviation  of  the  single  observation,  and 
will  be  denoted  by  a.d.  Thus 


This,  being  obviously  a  measure  of  the  deviations,  will  be 
called  the  deviation  measure  of  the  single  observation.  It  gives, 
at  least  approximately,  the  measure  of  the  variations  of  the 
resultant  indeterminate  errors  of  the  individual  observations. 
It  shows  also  that  in  the  given  series  the  observations  differ 
on  the  average  from  the  mean  by  this  amount ;  and  we  may 
infer  or  predict  that  more  observations  taken  under  the  same 
conditions  will  on  the  average  differ  from  this  mean  by  about 
this  amount. 

If  we  have  two  series  of  observations  consisting  of  a 
different  number  of  observations  nl  and  n9 ,  respectively,  all 
taken  under  the  same  conditions  and  with  equal  care,  then  the 
mean  result  of  the  series  for  which  n  is  greater  will  be  more 
free  from  the  effects  of  the  variable  parts  of  the  errors.  The 
principle  of  least  squares,  based  upon  the  general  law  of  de- 
viations, shows  that  the  reliability  in  this  respect  will  be  in 
proportion  to  the  square  roots  of  the  number  of  observations 
respectively,  that  is,  as  Vnl  :  Vn^.  Hence  we  may  say  that 
the  mean  result  of  a  series  of  observations  all  made  under  the 
same  conditions  and  with  equal  care  is  more  free  from  the 
effect  of  the  variable  parts  of  its  errors  in  proportion  to  Vn, 
that  is,  to  the  square  root  of  the  number  of  single  observations 


1 8  DIRECT  MEASUREMENTS. 

from  which  it  is  computed.  Hence  the  deviation  measure  of 
a  mean  result  would  be  that  of  the  single  observation  divided 
by  V~n.  Thus,  using  the  average  deviation,  the  deviation 
measure  of  the  mean  result  would  be 


a.d. 

—=  = — — [2] 

Vn       nVn 


This  will  be  called  the  Average  Deviation  of  the  mean.  It 
measures  the  effect  upon  the  mean  result  of  the  average  of 
the  variable  parts  of  the  errors  entering  into  the  single  ob- 
servations, and  obviously  bears  the  same  relation  to  a  mean 
result  that  a.d.  does  to  a  single  observation. 

Example  II. — Suppose  9  separate  observations  were  taken 
of  the  distance  between  two  points  with  the  results  headed  a  in 
the  table.  The  mean  result  to  be  used  would  then  be  A  = 
16.2799.  The  deviations  would  be  found  by  subtracting  A  from 
the  values  in  column  a,  and  are  given  in  column  d.  The  deviation 


9)  -269  9)  -°43 

A  =  16.2299  0.0048  =  ad. 


A  _         0.0048  . 

.\A.D.=    — 7?—     SB     o.ooi6cm. 


EXAMPLE  II.  19 

measure  of  the  single  observation  would  be  the  a.d.  =  0.0048. 
The  deviation  measure  of  the  mean  would  be  the  A.D.  = 
0.0016.  This  would  show  us  that  if  we  made  use  of  the  mean 
result  16.2299  in  any  work,  the  deviation  measure  to  be  used 
would  be  0.0016.  But  if  at  any  other  time  a  single  observa- 
tion only  were  made  of  the  same  distance  under  apparently 
identical  conditions,  and  that  single  result  were  to  be  used, 
the  deviation  measure  which  must  be  used  in  connection  with 
it  would  be  the  a.d.,  viz.  0.0048. 

The  relative  significance  of  the  a.d.  and  A.D.  may  be  put 
in  another  way  also.  If  we  wished  to  compare,  as  to  concord- 
ance, a  number  of  mean  results  taken  at  different  times  but 
under  similar  conditions  except  as  to  number  of  observations, 
we  should  use  the  A.D.  of  each  mean.  If  we  were  comparing 
the  relative  precision  of  the  single  observation  in  one  of  these 
series  with  that  in  any  other  one  we  should  make  use  of  the 


The  abbreviation  d.m.  will  be  occasionally  written  instead 
of  the  full  term  "  deviation  measure."  It  will  be  understood 
to  denote  any  deviation  measure,  viz.  the  a.d.,  A.D.,  or  any  of 
those  described  below,  according  to  the  context. 

The  deviation  measure  is  often  called  the  "  precision  meas- 
ure," *  but  the  latter  term  is  reserved  for  another  use  in  these 
pages. 

It  is  essential  to  note  exactly  the  significance  and  limita- 
tions of  the  deviation  measure.  It  does  not  tell  us  that  the 
result,  whether  a  single  observation  or  a  mean,  is  in  error  by 
this  stated  amount  (e.g.  the  a.d.  or  A.D.),  but  merely  that  the 
'variable  parts  of  the  errors  produce  a  variation  of  that  average 
amount  in  the  results.  By  the  law  of  distribution  of  these 
deviations  we  know  that  the  deviation  of  any  individual 
observation  may  be  many  times  the  a.d.\  or  of  a  mean  result, 
many  times  its  A.D.  In  fact  that  law  shows  that  the  chances 


*  This  usage  was  adhered  to  in  the  printed  Lecture  Notes  prepared  upon 
this  subject,  but  experience  has  shown  that  the  change  to  deviation  measure 
!•  desirable. 


20  DIRECT  MEASUREMENTS. 

that  the  deviation  will  assume  certain  specified  magnitudes  are 
those  given  in  this  table. 

o  100. 

J  a.d.  69. 

a.d.  43. 

2  a.d.  ii. 

3  a.d.  2. 

4  a.d.  O.  I 

Column  second  gives  the  percentage  of  the  whole  number 
of  observations  which  would  have  a  deviation  greater  than 
£  a.d.t  a.d,  2  a.d.,  etc.  Thus  in  any  series  sufficiently  large  to 
fulfil  the  conditions  under  which  the  general  law  of  deviations 
holds,  43  per  cent  of  the  single  observations  would  have  a 
deviation  greater  than  the  average,  n  per  cent  only  (i.e. 
about  one  in  ten)  greater  than  twice  the  average,  and  only  one 
in  one  thousand  greater  than  four  times  the  average.  Thus  in 
the  foregoing  example,  where  the  a.d.  was  0.0048,  we  may  say 
that  the  chances  are  43  to  57,  or  roughly  about  even,  that  any 
single  observation  is  affected  by  the  variable  parts  of  the  errors 
to  an  extent  of  ±  0.0048  units.  The  A.D.  of  the  mean  of  that 
series  is  0.0016,  so  that  we  may  say  of  the  mean  that  the 
chances  are  nearly  even  that  it  is  thus  affected  to  the  extent 
of  about  ±  0.0016  units. 

Places  of  Figures  in  d.tn.;  and  Negligible  Amounts. — In 
the  numerical  value  of  any  deviation  measure,  two  and  only 
two  significant  figures  should  be  retained  ;  as  was  done  in  the 
above  example.  Any  single  change  in  the  measured  quantity, 
a,  due  to  whatever  cause  may  be  regarded  as  negligible  when 
not  exceeding  TVth  of  the  deviation  measure  of  the  quantity. 
Therefore  a  should  be  carried  out  to  the  place  correspond- 
ing to  the  last  significant  figure  of  the  d.m.  Similarly  any 
change  in  the  d.m.  is  negligible  when  not  exceeding  ^d.m. 

The  fractional  and  percentage  deviations,  d.m. /a  and  100 
d.m./a  (see  page  29),  should  also  contain  two  and  only  two 
significant  figures ;  and  any  fractional  change  is  negligible  in 


PLACES  OF  FIGURES  IN  d.m.  21 


them  when  not  exceeding  ^th  of  their  values.  Similarly  any 
fractional  change  in  the  measured  quantity  a  is  negligible 
when  not  exceeding  ^  d.m./  a. 

These  statements  may  be  justified  as  follows  :  Taking  the 
numbers  used  in  the  above  example,  let  16.2299  denote  a  mean 
result  of  a  direct  measurement,  and  0.0016  its  deviation  meas- 
ure. The  latter  shows  that  the  number  16.2299  *s  uncertain 
by  16  units  in  the  sixth  place  of  significant  figures.  A  change 
corresponding  to  -fad.m.  would  be  2  in  this  sixth  place,  already 
uncertain  by  16.  It  is  therefore  clear  that  such  a  change  is 
immaterial  and  may  be  regarded  as  negligible.  This  change 
of  2  being  negligible  in  the  number  an  equal  change  would  be 
negligible  in  the  d.m.,  and,  as  this  is  10  per  cent,  of  that  num- 
ber, a  change  in  d.m.  of  -^d.m.  is  negligible.  Obviously  also 
the  figure  corresponding  to  -^d.m.  will  always  be  in  the  second 
place  of  significant  figures,  so  that  if  we  always  retain  that 
place  and  always  reject  all  figures  beyond  that  place  in  d.m., 
we  shall  never  introduce  by  that  process  an  error  exceeding 
this  limit  into  the  d.m.  Hence  two  places  of  significant  figures 
in  d.m.  are  enough. 

This  limit  of  -fad.m.  as  the  negligible  amount  is  an  arbi- 
trary selection.  A  larger  or  a  smaller  amount  might  have  been 
chosen  as  the  limit,  but  experience  shows  this  to  be  both  con- 
venient and  suitable  in  practice.  Yet  in  rather  rough  work  a 
larger  limit  may  be  used,  and  for  such  work  the  d.m.  need  be 
retained  only  to  one  place  when  not  less  than  5  in  that  place. 
For  instance,  if  the  above  example  represented  rather  rough 
work,  the  a.d.  might  be  written  0.005  instead  of  0.0048,  but 
the  A.D.  would  rarely  be  written  0.002  instead  of  0.0016. 

By  inspection  it  is  easy  to  see  from  these  statements  that 
the  numerical  result  should  in  general  be  carried  out  to  the 
place  corresponding  to  that  of  the  second  significant  figure  of 
the  d.m.  of  the  quantity.  Thus  the  number  16.2299  should  be 
carried  out  to  the  sixth  place  of  figures.  This  statement  is 
true  whether  the  result  is  a  mean  or  a  single  observation,  the 
d.m.  being  in  the  first  case  the  A.D.,  in  the  second  the  a.d.  In- 
spection of  the  data  in  any  case  will  usually  show  us  what  place 


22  DIRECT  MEASUREMENTS. 

will  correspond  to  the  second  of  the  d.m.  even  in  advance  of 
the  exact  computation  of  that  quantity. 

It  is  obvious  that  if  -f^d.m.  is  negligible,  ^d.m./a  will  be 
also,  for  both  are  the  same  part  of  a.  Similarly  if  d.m.  must 
be  carried  to  two  places  to  correspond  to  this  limit,  ^d.m./a 
must  also  be  carried  to  two  places.  The  same  is,  of  course, 
true  for  the  percentage  precision. 

In  computing  the  deviation  d  by  subtracting  A  from  a1 , 
etc.,  it  is  usually  unnecessary  to  retain  for  this  part  of  the 
work  more  places  in  A  than  are  given  in  al9  a^,  etc.,  in  the 
observations.  Thus  in  the  foregoing  example,  to  find  dl ,  etc., 
we  use  16.230  instead  of  16.2299.  H*  however,  the  values  of  d 
are  very  small,  the  largest  value  not  exceeding  perhaps  2  or  3 
units  in  the  last  place,  then  it  is  better  to  retain  the  full  num- 
ber of  places  in  A  or  at  least  one  more  than  in  the  values  of  a. 
For  instance  in  the  example  if  16.233  nad  been  the  largest  and 
16.228  the  smallest  value  of  a  and  the  mean  had  been  16.22  993 
the  deviations  would  have  been  formed  using  16.2299.  It  would 
be  useless, however, to  retain  16.22993  for  this  purpose, although 
it  might  be  proper  to  retain  it  for  other  uses.  It  is,  however, 
to  be  noted  that  when  the  apparatus  gives  indications  which 
continually  agree  within  one  or  two  units  in  the  last  place  of 
figures  obtained  by  the  single  observation,  it  is  delusive  to 
hope  for  much  gain  in  precision  by  many  repetitions.  Such 
cases  often  occur  in  practice.  They  usually  show  that  the 
indicating  part  of  the  apparatus,  whatever  it  may  be,  is  not  as 
sensitive  as  it  might  advantageously  be  made.  Thus  if  in 
making  a  weighing  of  the  same  object  repeatedly  by  the  ordi- 
nary method,  we  find  that  the  results  agree  to  one  or  two  units 
in  the  last  place,  e.g.  to  o.i  or  0.2  mgr.,  this  indicates  that 
the  balance  is  delicate  enough  to  have  a  finer  index  or  to  be 
used  by  the  method  of  swings. 

Best  Value  of  n. — The  question  continually  arises,  how 
many  observations  is  it  worth  while  to  take  in  order  to  reduce 
theA.D.  of  the  mean  ?  Since  A.D.  =  a.d./  Vn,  the  gain  is  only 
in  proportion  to  Vn.  But  the  labor  of  observing  is  in  direct 
proportion  to  n.  Thus  to  double  the  gain  the  labor  must  be 


OTHER  DEVIATION  MEASURES.  2$ 

fourfold,  to  treble  it  ninefold,  i.e.  the  labor  is  as  the  square  of 
the  gain.  Obviously  then  a  point  would  soon  be  reached  where 
the  labor  would  become  excessive  in  comparison  with  the  gain 
or  with  the  labor  involved  in  other  parts  of  the  work.  The 
limit  to  the  number  n  of  observations  to  be  taken  must  then 
be  determined  by  the  judgment  of  the  observer  as  to  when  the 
labor  becomes  excessive  in  proportion  to  the  gain.  In  ordi- 
nary work  n  =  9  is  often  a  convenient  and  sufficient  number, 
though  a  smaller  number  will  frequently  suffice.  It  is  rare 
except  in  the  most  careful  work,  or  in  work  of  some  special 
character,  that  n  is  made  to  exceed  25. 

Other  Deviation  Measures.  —  Other  quantities  than  the 
average  deviation  are  also  employed  as  deviation  measures. 
In  fact  the  most  common  measure  is  not  the  average  deviation 
but  the  so-called  "  probable  error."  The  relation  between  the 
probable  error  and  average  deviation  is  given  by  the  expressions 


p.e.  =  0.84  a.d.  ;     P.E.  =  o.fyA.D.,  .     .     .     .     [3] 

where  p.e.  is  the  probable  error  of  the  single  observation,  and 
P.E.  that  of  the  mean  result.  The  ordinary  formulae  for  com- 
puting the  probable  errors  from  the  square  root  of  the  sum  of 
the  squares  of  the  deviations  possess  no  real  advantages  over 
the  above,  while  far  more  laborious.  The  probable  error,  p.e.,  is 
merely  a  deviation  of  such  magnitude  that  there  are,  in  a  large 
series,  just  as  many  deviations  greater  as  less  than  it  ;  or  in 
other  words,  such  that  in  the  series  there  are  just  as  many 
observations  having  values  lying  between^  -\-p.e.  and  A  —p.e. 
as  outside  those  limits  ;  so  that  it  is  an  even  chance  whether 
any  observation  taken  at  random  will  have  a  deviation  greater 
or  less  than  p.e. 

The  use  of  the  "  probable  error  "  is  objectionable  partly 
because  of  its  more  artificial  character,  partly  because  of  the 
greater  labor  of  computation,  but  chiefly  because  the  term  is 
seriously  misleading.  It  is,  in  the  first  place,  not  an  error  at 
all,  but  merely  a  deviation.  Neither  is  it  a  "probable"  value 
in  the  ordinary  sense,  as  it  is  more  probable,  that  is  of  greater 


24  DIRECT  MEASUREMENTS. 

frequency  of  occurrence,  than  any  given  larger  deviation,  and 
less  probable  than  any  smaller  one.  Its  use  leads  almost 
inevitably  to  a  fallacious  impression  as  to  the  real  accuracy  of 
results,  and  tends  to  promote  negligence  as  to  the  constant 
errors  which  are  of  far  more  serious  importance.  For  a  reader 
meeting  a  result  stated  to  have  a  small  probable  error  is  liable, 
unless  unusually  upon  his  guard,  to  receive  at  once  an  impres- 
sion of  accurate  work  and  to  have  his  attention  diverted  from 
other  points  upon  which  the  reliability  of  the  work  depends  to  a 
greater  extent.  And  an  observer,  with  the  natural  tendency  to 
confidence  in  his  own  work,  is  even  more  easily  misled  by  the 
term  "probable  error."  The  term  "average  deviation"  tends, 
on  the  contrary,  rather  to  call  attention  to  the  true  character 
of  the  quantity;  and  by  the  use  of  "error"  solely  in  connec- 
tion with  constant  errors,  attention  is  the  more  strongly  directed 
upon  these.  To  a  competent  and  experienced  observer  this 
discrimination  in  terms  is  unimportant,  but  it  is  by  no  means 
so  to  the  beginner. 

It  may  be  remarked  that  the  numerical  difference  between 
the  average  deviation  and  the  probable  error  is  negligible  in 
almost  all  work  if  we  follow  the  limit  already  set  (viz.,  y1^  a.d. 
or  y\j-  A.D.).  It  is  also  of  course  true  that  in  all  the  formulae 
developed,  the  probable  error  may  be  inserted  to  replace  the 
average  deviation,  if  desired,  a  little  attention  being  given  to 
insure  consistency. 

Special  Laws  of  Deviations. — Besides  the  foregoing  general 
law  there  are  other  laws  which  the  deviations  follow  in  certain 
cases.  Of  these  special  laws  the  only  one  with  which  we  are 
concerned  is  that  occurring  when  any  deviation  between  the 
limits  -f  a  and  —  a  is  equally  likely  to  be  obtained,  i.e.,  where  all 
deviations  between  these  limits  have  an  equal  frequency.  It 
is  easy  to  see  by  inspection  that  the  average  deviation  under 
this  law  must  be  \a. 

This  is  the  law  according  to  which  the  deviations  occur 
when  tenths  of  a  division  are  estimated  by  the  eye.  With 
moderate  practice,  divisions  of  not  less  t'han  half  a  millimeter 
can  be  read  to  tenths  with  the  unaided  eye  so  that  the  estima- 


PRECISION  MEASURE   OF  RESULT.  2$ 

tion  shall  always  give  the  nearest  tenth,  that  is,  so  that  the 
error  or  deviation  shall  not  exceed  -|-  0.05  or  —  0.05  mm. 
Now  as  the  point  to  be  read  is  equally  likely  to  lie  anywhere 
along  the  scale,  its  actual  distance  from  the  estimated  tenth  is 
equally  likely  to  be  anything  within  these  limits.  Thus  the 
a.d.  of  a  single  estimation  will  be  0.025  or  -fath  of  a  division. 
Experience  demonstrates  that  this  limit  is  reached  without 
difficulty,  and  often  exceeded  where  an  attempt  is  made  under 
good  conditions  to  estimate  twentieths  instead  of  tenths. 

Precision  Measure  of  Result.  —  Let  d.m.  denote  the  devia- 
tion measure  of  the  result,  viz.  the  a.d.  if  the  result  be  a  single 
observation,  and  A.D.  if  it  be  a  mean.  Let  r  denote  the 
measure  of  a  residual  (page  n)  left  by  the  elimination  of  a 
determinate  error,  r,,ra,...,  rp  being  the  measure  of  the 
respective  residuals  from/  determinate  errors. 

Then  the  term  precision  measure,  p.m.,  will  be  hereafter 
used  to  denote  the  quantity  6  given  by  the  expression 


The  precision  measure  of  a  direct  result  includes  therefore  both 
the  deviation  measure  and  the  residual  measures  of  all  deter- 
minate errors,  so  far  as  they  can  be  numerically  expressed.  It  is 
thus  the  best  and  only  numerical  measure  obtainable  of  the  accu- 
racy of  that  result  taken  by  itself,  but  it  fails,  of  course,  to  in- 
dicate anything  more  respecting  the  constant  errors  than  to 
imply  that  so  far  as  determinate  these  have  been  removed. 

If  the  residuals  r  are  all  negligible  as  compared  with  the 
d.m.y  then  the  precision  and  deviation  measures  coincide.  The 
law  of  accumulation  by  squares  from  which  the  above  expres- 
sion for  d  is  deduced  is  based  on  the  principle  of  least  squares, 
and  will  be  further  discussed  in  late  sections. 

The  term  precision  is  used  intentionally  rather  than  accu-- 
racy  in  the  foregoing  paragraphs.  A  distinction  between  the 
denotation  of  these  terms  will  be  maintained.  Accuracy  will 
be  used  only  when  attention  is  distinctly  directed  toward  the 
constant  as  well  as  the  variable  errors.  An  estimate  of  the 


26  DIRECT  MEASUREMENTS. 

accuracy  of  a  result  thus  involves  a  discussion  of  possible  con- 
stant errors.  The  precision  measure  although  implying  when 
properly  used  that  no  determinate  constant  errors  remain,  does 
not  call  for  a  discussion  of  the  constant  errors.  A  result  might 
be  precise,  and  yet  contain  a  large  unknown  constant  error,  but  it 
would  not  then  be  accurate,  in  the  sense  in  which  these  terms 
are  here  employed. 

To  Make  Residuals  Negligible  in  P.M. — One  or  more  of 
the  residuals,  r,  in  the  expression  i^  p.m.  may  become  negligible. 

Criterion. — The  criterion  is  as  follows  : 

I.  Any  single  residual,  r,  is  negligible  when 

r  <  fan [5] 


2.  Any  number  q  of  residuals  is  simultaneously  negligible 
when 


where  2r*  is  the  sum  of  the  squares  of  the  neglected  residuals. 
A  convenient  though  less  broad  form  of  this  second  criterion 
is  that  each  neglected  residual,  r,  shall  be 

r  <  \p.m./  Vq [7] 

This  last  is  based  on  the  assignment  of  equal  effects  discussed 
in  the  next  section. 

Note. — In  the  expressions  [5],  [6],  and  [7]  we  may  sub- 
stitute for  p.m.  either  p.m.1  or  d.m.,  where/.**.1  is  the  precision 
measure  when  r2  or  2r*  is  neglected,  and  d.m.  is  the  deviation 
measure  from  which  p.m.  is  computed.  The  limit  thus  fixed 
is  smaller  than  that  given  by  [5],  [6],  and  [7],  because  p.m.1 
and  d.m.  are  smaller  than  p.m. 

This  change  in  form  is  convenient  in  many  actual  cases, 


BEST   VALUE   OF   THE  RESIDUALS;  EQUAL  EFFECTS.     2J 

because  we  may  frequently  know  d.m.  or  even  p.m.1  where  we 
cannot  equally  well  find/.w.  or  even/.w.1 

Demonstration.  —  Let  S  represent  the  p.m.,  including  r, 
and  6l  that  omitting  r.  Then  for  the  greatest  value  of  r 
negligible 


But 


Hence 


.%    r  =  0.44$. 

Hence  r  would  be  negligible  if  as  great  as  0.44^,  but  a  limit 
of  0.33$  or  -Jtf  will  be  here  assumed,  partly  as  being  a  con- 
venient value,  partly  as  being  rather  safer  than  0.44^  where  as 
usual  the  number  of  residuals  is  small  so  that  the  assumed 
formula  of  squares  is  less  exact. 

The  effect  on  p.m.  of  any  group  of  q  residuals  is  2r*.  This 
group  would  obviously  be  negligible  under  the  same  condition 
which  made  r  so  in  the  first  case. 

This  limit  at  which  the  effect  of  the  residuals  becomes 
negligible  is  unfortunately  beyond  attainment  in  many  cases 
in  practice.  With  a  considerable  proportion  of  all  direct  read- 
ing instruments  the  corrections  cannot  be  determined  much 
more  closely  than  the  a.d.  of  the  ordinary  readings. 

Best  Value  of  the  Residuals  :  Equal  Effects.  —  The  follow- 
ing case  is  of  frequent  occurrence.  Given  a  direct  measure- 
ment in  which  there  are  /  residuals  not  negligible  but  whose 


28  DIRECT  MEASUREMENTS. 

joint  effect  must  not  exceed  a  stated  limit  /;  to  what  value  or 
limit  is  it  most  advantageous  to  determine  each  residual. 
This  is  essentially  the  same  problem  that  is  discussed  fully 
later  in  a  section  headed  "  Equal  Effects  "  for  the  components 
of  an  indirect  measurement.  Only  the  result  therefore  will  be 
here  stated.  The  relation  of  the  residuals  to  /  is 


The  best  value  for  the  residuals  is  given  approximately  by 


that  is,  they  must  all  be  equal  and  therefore  of  "  equal  effect  " 
on  /and  thus  on  p.m. 

Although  this  rule  affords  the  best  solution  to  use  as  a 
starting  point,  it  is  only  approximate  and  not  necessarily  final. 
The  exact  values  of  r  which  would  be  best  in  every  case  would, 
of  course,  be  those  which  would  give  the  stated  value  of  /  with 
the  least  labor.  We  cannot,  however,  readily  obtain  a  solution 
on  this  basis.  The  values  given  by  (11)  will  comply  with  this 
requirement  only  when  there  is  equal  difficulty  in  obtaining 
each  of  them.  If  some  of  the  eliminations  are  more  difficult 
than  others,  then  the  best  values  for  the  residuals  of  the  more 
difficult  would  be  larger,  those  for  the  less  difficult  being  there- 
fore correspondingly  smaller  ;  but  the  departure  from  equality 
must  always  be  small.  The  residuals  of  the  more  difficult 
eliminations  should  rarely  be  allowed  to  increase  to  twice  the 
value  for  equal  effects,  and  if  one  or  more  of  the  residuals  is 
increased  the  others  must  be  correspondingly  diminished  in 
order  that  the  limiting  value  of  /  shall  not  be  exceeded.  It 
must  be  left  largely  to  the  judgment  of  the  observer  to  deter- 
mine what  departure  shall  be  made  from  the  condition  of  equal 
effects.  The  further  statement  made  in  the  paragraph  referred 
to  should  be  consulted. 

The  usual  case  is  where  the  result  is  desired  with  a  stated 
p.m.,  and  the  d.m.  of  the  apparatus  is  fixed  or  is  determinable 


FRACTIONAL  DEVIATION.      FRACTIONAL  PRECISION.      2g 

by  a  preliminary  trial.  The  numerical  value  of  the  limit  /  of 
the  combined  effect  of  the  residuals  will  then  usually  be  de- 
termined by  the  expression 

l*=p.m*  —  d.m? -.     . 

In  case  the  d.m.  is  not  known  in  advance  it  must  be  found  at 
the  outset  by  making  a  preliminary  series  of  observations. 
Now  the  final  result  will  generally  be  a  mean,  so  that  its  d.m. 
will  be  the  A. D.  To  find  this  from  the  preliminary  observa- 
tions, we  must  know  the  number  n  of  observations  which  are  to 
enter  into  the  final  mean.  But  this  cannot  be  fixed  upon  at 
this  stage,  so  that  it  is  necessary  to  assume  a  number.  Ordi- 
narily it  will  be  on  the  safe  side  to  assume  n  =  25,  and  thus 
find  d.m.  —  A.D.  =  a.d./  V2$  from  the  preliminary  value  of 
a.d.  If  this  cannot  be  done  it  is  sometimes  sufficient  to  esti- 
mate a  value  of  d.m.  for  a  rough  preliminary  calculation  of  the 
limits  of  r. 

Fractional  Deviation.  Fractional  Precision. — Let  a  de- 
note a  single  observation  and  a.d.  its  deviation  measure  ;  then 
a.d./a  is  the  fractional  deviation — more  properly  fractional 
deviation  measure — of  the  single  observation.  Similarly 
100  a.d./a  is  the  percentage  deviation.  If  A  be  the  mean  of  a 
series  of  values  of  a,  and  A.D.  its  deviation  measure;  then 
A.D./A  will  be  the  fractional  deviation  of  the  mean,  and 
100  A.D./A,  the  percentage  deviation.  These  quantities  are 
to  be  carried  to  two  places  of  significant  figures  only  since  a.d. 
and  A.D.  are  so,  for  reasons  already  stated  Therefore  a  very 
rough  value  (two  places  of  significant  figures)  is  all  that  is 
needed  for  a  or  A, — an  important  point  in  some  applications 
of  the  methods  of  this  subject.  For  this  reason  we  obviously 
may  substitute  A  for  a  in  the  above  formula. 

Similarly  also  if  p.m.  is  the  precision  measure  of  any 
quantity  a  whether  a  mean  or  a  single  observation,  p.m./a  is 
its  fractional  precision  and  100  p.m./a  is  its  percentage  pre- 
cision. The  above  remarks  as  to  significant  figures  also  apply- 
here. 


3O  DIRECT  MEASUREMENTS. 

Mistakes. — Errors  due  to  such  causes  as  recording  an 
observed  number  incorrectly,  counting  up  weights  wrongly,  in- 
correct numbering  of  a  scale,  faulty  arithmetical  work,  are 
classed  as  mistakes.  In  any  work  under  ordinary  conditions 
where  the  number  of  observations  is  not  great,  these  mistakes 
do  not  fall  in  with  the  deviations,  and  are  not  eliminated  by 
averaging.  They  can  only  be  detected  by  careful  inspection 
and  by  check  observations  or  computations. 

Criterion  for  Rejection  of  Doubtful  Observations. — It 
often  happens  that  among  several  single  observations  taken 
under  the  same  conditions  and  with  equal  care  one  deviates 
quite  widely  from  the  rest,  so  widely  that  the  possibility  of  its 
containing  some  mistake  suggests  itself.  In  such  a  case,  in- 
spection of  the  records  or  of  the  work  may  show  conclusively 
that  a  mistake  did  occur,  and  may  possibly  point  out  its  exact 
amount.  If  the  existence  of  the  mistake  is  thus  established, 
the  faulty  observation  should  in  general  be  cancelled  and 
wholly  rejected  ;  but  if  inspection  shows  positively  its  exact 
amount,  the  mistake  may  be  rectified.  It  is  always  better  to 
reject  an  observation  than  to  correct  it  unless  the  mistake  is 
perfectly  obvious  and  its  amount  certain  beyond  question. 
Usually,  however,  the  cause  of  the  wide  deviation  is  not 
apparent,  no  obvious  mistake  being  discoverable,  and  the 
question  arises  as  to  whether  the  observation  should  then  be 
rejected  or  retained.  Mathematical  criteria  based  on  the 
theory  of  probabilities  have  been  given  by  Peirce,  Chauvenet, 
and  others  to  decide  this  question,  in  any  given  case,  but 
they  are  somewhat  complicated  in  application,  and  a  much 
simpler  one  is  sufficient  for  ordinary  work. 

Criterion. — Take  the  mean  and  the  a.d.  of  the  observations, 
omitting  the  doubtful  one.  Find  the  deviation,  d.y  of  that  one 
from  the  mean.  Then  reject  the  observation  if  d  >  ^a.d. 

This  limit  is  arbitrary  and  might  perhaps  be  made  narrower 
to  advantage.  It  is  not  based  upon  any  supposition  that  an 
observation  with  a  greater  deviation  than  ^a.d.  necessarily  or 
even  presumably  contains  a  mistake.  On  the  contrary,  if  the 
law  of  deviations  is  followed,  observations  with  a  greater  de- 


WEIGHTS.  31 

viation  than  this  will  sometimes  although  infrequently  occur,  the 
frequency  of  deviations  greater  than  ^.a.d.  being  only  I  in  1000. 
The  basis  of  any  such  criterion  is  rather  this  :  That  inasmuch 
as  the  number  of  observations  in  the  series  is  always  compara- 
tively small,  the  large  infrequent  deviation  would  have  undue 
influence  if  allowed  to  remain ;  so  that  the  mean  taken  after 
rejecting  it  is  likely  to  be  more  reliable  than  that  which  would 
result  if  it  were  retained. 

Something  must  also  be  left  to  the  judgment  of  the  ob- 
server as  to  the  propriety  of  making  a  rejection ;  and  he  is 
especially  entitled  to  exercise  an  autocratic  power  in  this  re- 
gard if  he  has  good  reason  for  eveft  suspecting  that  some  ex- 
cessive or  extraordinary  cause  of  error  has  influenced  any 
given  observation.  If  this  is  the  case  the  observation  ought 
invariably  to  be  rejected,  for  one  doubtful  observation  may 
vitiate  a  mean  by  a  greater  amount  than  can  be  compensated 
by  many  good  ones. 

There  is  a  tendency,  especially  among  inexperienced  ob- 
servers, to  become  biassed  by  the  first  one  or  two  readings  of 
a  series,  and  to  reject,  without  recording  it,  any  later  one  which 
does  not  closely  accord  with  these,  tacitly  assuming  it  to  be 
faulty.  This  is  an  essentially  vicious  practice  which  cannot  be 
too  carefully  avoided.  Other  things  being  equal  the  later  ob- 
servations are  entitled  to  greater  rather  than  less  weight  than 
the  earlier  ones,  and  no  result  should  be  rejected  without 
sufficient  warrant.  Above  all  things,  the  integrity  of  the 
observer  must  be  beyond  question  if  he  would  have  his 
results  carry  any  weight,  and  it  is  in  the  matter  of  the  rejec- 
tion of  doubtful  or  discordant  observations  that  his  integrity 
in  scientific  or  technical  work  meets  its  first  test.  It  is  of 
hardly  less  importance  that  he  should  be  as  far  as  possible  free 
irom  bias  due  either  to  preconceived  opinions  or  to  uncon- 
scious efforts  to  obtain  concordant  results. 

Weights. — Suppose  several  different  independent  measure- 
ments (e.g.  by  different  methods,  observers,  etc.),  to  have 
been  made  of  the  same  quantity.  Let  al ,  #2 ,  .  .  .  an  denote 
the  results,  and  p.m.^  p.m.^,  ...p.m.n  their  respective  pre- 


32  DIRECT  MEASUREMENTS. 

cision  measures.  And  suppose  further  that  it  is  desired  to 
find  from  these  results  the  best  representative  value.  Then  if 
these  precision  measures  give  us  proper  indications  of  the  re- 
liability of  the  results,  that  is  if  in  each  case  the  constant 
error,  so  far  as  discoverable,  is  negligible  compared  with  the 
p.m.,  the  weight/  to  be  assigned  to  each  result  is  inversely  as 
the  square  of  itsfl.m.  Thus 


The  best  representative  value  will  then  be  the  weighted  mean 
viz., 


A+A  +  -. 

The  demonstration  of  this  proposition  is  given  in  treatises  on 
Least  Squares. 

Meaning  of  Estimated  Accuracy  of  Direct  Result. — This 
can  now  be  readily  defined.  When  we  estimate  the  accuracy 
of  a  result  of  a  direct  measurement  at  a  given  amount  (e.g.,  if 
we  say  that  it  appears  correct  to  2  per  cent.),  we  mean  that 
the  precision  measure  of  the  result  does  not  exceed  that 
amount,  and  that  so  far  as  we  can  discover  there  is  no  con- 
stant error  which  is  sensible  (i.e.  not  negligible)  compared 
with  this/.w. 

We  do  not  mean  that  the  actual  error  of  the  result  is  of 
just  this  amount,  for  if  we  did  we  should  correct  accordingly. 
Neither  do  we  mean  that  this  is  a  more  probable  value  of  the 
error  than  any  other.  But  using  the  average  deviation  as  the 
d.m.,  we  mean  that  the  average  effect  of  all  the  errors  remain- 
ing, so  far  as  we  can  discover,  is  of  this  amount  and  may  be 
either  -\-or-  in  sign.  This  implies  that  if  several  results  of 
this  kind  were  to  be  obtained  under  the  same  conditions,  the 
average  discrepancy  among  them  would  be  approximately  of 
this  amount. 

Similarly  when  we  say  that  a  result  is  desired  with  an  accu- 


FORMS  OF  PROBLEMS  ON  ACCURACY  OF  RESULT.        33 

racy  of  a  stated  amount,  we  mean  that  the  measurement  is  to 
be  so  made  that  the  precision  measure  of  the  result  shall  not 
exceed  the  corresponding  amount ;  and  that  so  far  as  is  dis- 
coverable the  constant  error  shall  not  be  sensible  compared 
with  this. 

Forms  of  Problems  on  Accuracy  of  Result. — Concerning 
the  accuracy  of  the  result  of  a  direct  measurement  by  any 
single  method,  problems  arise  in  three  different  forms. 

First.  To  obtain  by  a  proposed  method  the  most  accurate 
result  practicable. 

Second.  To  obtain  a  direct  measurement  of  a  desired  quan- 
tity and  have  the  result  accurate  within  a  specified  limit. 

Third.  Given  a  completed  result  obtained  by  a  stated 
method  to  estimate  its  accuracy. 

First.  To  obtain  by  a  proposed  method  the  most  accurate 
result  practicable.  To  accomplish  this  the  elimination  of 
errors  must  be  carried  as  far  as  practicable,  i.e.  as  far  as  the 
conditions  and  the  amount  of  labor  which  can  be  devoted  to 
the  work  will  permit.  Thus  all  constant  errors  as  well  as  the 
deviation  measure  must  be  reduced  to  the  smallest  practicable 
limit. 

For  this  purpose,  the  method,  apparatus,  and  conditions  of 
work  must  be  thoroughly  studied  to  discover,  as  far  as  possi- 
ble, all  sources  of  error,  with  a  view  to  their  removal  or  to  the 
elimination  of  their  effects.  As  many  as  possible  of  these 
sources  must  then  be  removed  by  modifying  the  method, 
apparatus,  or  conditions  of  working.  The  magnitude  of  the 
effects  of  the  remaining  determinate  sources  of  error  must 
then  be  evaluated,  i.e.  corrections  determined  for  them. 
Finally,  a  series  of  observations  must  be  taken  so  that  their 
average  may  reduce  the  effect  of  the  variable  indeterminate 
errors. 

To  make  the  result  the  most  accurate  practicable  with  the 
method,  the  removals  and  corrections  must  be  made  with  suffi- 
cient exactness  to  reduce  their  residuals  to  negligible  amounts, 
or  rather  this  limit  must  be  approached  as  closely  as  can  be 
done  without  excessive  labor.  Also  the  observations  must  be 


34  DIRECT  MEASUREMENTS. 

as  numerous  as  may  be  without  undue  labor.  The  resulting 
precision  measure  and  also  the  constant  error  of  the  result 
will  then  be  as  small  as  it  is  practicable  to  make  them,  and  the 
result  will  therefore  be  as  accurate  as  it  is  practicable  to 
obtain  by  the  method. 

The  limit  as  to  what  is  "  practicable  "  is  determined  by  the 
labor  involved  in  various  parts  of  the  work.  What  the  limit 
shall  be  which  may  be  regarded  as  excessive  in  the  reduction 
of  the  residuals  to  a  negligible  amount,  must  be  largely  left  to 
the  judgment  of  the  observer.  Obviously,  however,  it  is  not 
to  be  determined  by  the  limit  of  equal  labor  on  each  removal. 
This  amount  on  the  different  removals  will  differ  widely.  It 
is  also  evident  that  where  the  removal  is  easy,  it  may  well  be 
made  with  a  little  greater  accuracy,  when  difficult  with  a 
little  less,  than  the  amount  corresponding  exactly  to  the 
negligible  limit  (page  26).  But  to  pass  far  beyond  the  limit 
on  either  side  involves  a  poor  distribution  of  the  labor,  and 
thus  a  less  accurate  result  than  might  be  reached  with  the 
same  amount  of  labor.  As  to  the  labor  to  be  expended  in 
repetition  of  observation,  this  repetition  is  often  so  simple  a 
matter  that  it  would  be  foolish  to  neglect  to  take  a  consider- 
able number  of  observations  to  materially  increase  the  precision 
of  the  result ;  but  on  the  other  hand  it  is  equally  unwise  to  con- 
tinue the  observations  beyond  the  point  at  which  the  labor  in- 
volved becomes  large  compared  with  that  in  the  rest  of  the 
work.  After  the  first  few  observations  the  gain  is  very  slow  in 
proportion  to  the  total  labor,  as  already  shown.  It  is  seldom 
worth  while  to  reduce  the  A.D.  below  the  residuals,  where 
these  exist,  unless  it  can  be  very  easily  done. 

Second.  To  obtain  a  direct  measurement  of  a  desired  quan- 
tity•,  and  have  the  result  accurate  within  a  specified  limit. 

We  must  for  this  end  fix  upon  a  method,  apparatus,  etc., 
which  will  give  the  result  with  a  precision  measure  not  exceed- 
ing the  specified  limit.  To  establish  the  accuracy  of  the  result 
it  is  further  necessary  to  show  that  the  sources  of  error  have 
been  so  well  studied  that  there  is  no  indication  that  the  con- 


FORMS  OF  PROBLEMS  ON  ACCURACY  OF  RESULT.       35 

stant  error  of  the  result  is  comparable  with  the  precision  meas- 
ure finally  attained. 

The  usual  order  of  procedure  is  to  find  the  d.m.  by  pre- 
liminary trial.  This  with  the  prescribed  precision  measure 
fixes  the  normal  values  of  the  residuals.  In  general,  as  stated 
at  page  28,  the  labor  will  be  distributed  nearly  to  the  best  ad- 
vantage when  rl  =  ra  =  ra  =  . , .  =  rp  =  //  Vn . 

The  method,  etc.,  would  then  be  studied  to  determine 
whether  this  limit  could  be  reached,  or  what  limit  would  be 
practicable.  Finally,  the  precision  measure  would  be  recom- 
puted from  the  practicable  values  of  the  residuals,  to  see 
whether  it  exceeded  the  prescribed  limit,  in  which  case  the 
method  must  be  modified  or  a  less  accurate  result  accepted. 

After  the  completion  of  the  actual  observations  it  is  of 
course  necessary  to  make  a  final  estimate  of  the  accuracy  of 
the  result. 

Third.  Given  a  completed  result  obtained  by  a  stated  method, 
to  estimate  its  accuracy. 

This  question  arises  in  reviewing  results  of  any  work  after 
its  completion.  A  statement  will  first  be  made  which  will 
apply  to  any  piece  of  work  done  by  another  person  and  com- 
ing before  us  for  inspection.  It  thus  will  apply  to  any  pub- 
lished work  of  quantitative  character.  From  what  has  been 
already  said  under  the  two  preceding  cases  we  can  at  once  see 
that  the  procedure  is  as  follows. 

Study  the  method,  apparatus,  and  conditions  to  discover 
as  far  as  possible  all  sources  of  error.  Ascertain  whether  these 
have  been  removed  or  corrected  for.  If  so  ascertain  the  meas- 
ure of  the  residual  from  each.  If  not,  all  the  unremoved 
sources  constitute  causes  of  constant  error  whose  magnitude 
must  be  determined  or  estimated  and  considered  in  the  final 
summing  up.  Lastly  the  deviation  measure  must  be  ascer- 
tained, and  from  this  and  the  residuals  the  precision  measure 
must  be  calculated.  Any  unremoved  constant  error  dis- 
covered may  be  treated  as  a  residual  in  the  calculation  of  the 
corresponding  precision  measure  if  its  amount  or  average  value 
can  be  estimated. 


36  DIRECT  MEASUREMENTS. 

Finally,  give  the  resulting  precision  measure,  and  state 
whether,  so  far  as  discoverable,  all  determinate  sources  of  error 
have  been  removed  or  their  effects  corrected  so  that  the  con- 
stant error  of  the  result  is  negligible  compared  with  the  p.m. 
This  forms  the  whole  "  estimate  of  the  accuracy"  of  the  result. 
It  is  usually  well  to  express  the  p.m.  both  in  units  and  in  per- 
centage. 

Data  Required  to  Substantiate  Result. — In  order  to  de- 
termine the  reliability  and  value  of  a  result  it  is  necessary  to 
form  such  an  estimate  of  its  accuracy.  To  do  this  it  is  essen- 
tial, of  course,  that  all  the  necessary  data  should  be  in  hand. 
Any  description  of  a  method  and  result,  as  in  a  published 
paper,  can  then  be  justly  criticised  as  materially  incomplete 
if  it  does  not  give  all  the  data  needed  for  such  a  discussion. 
The  importance  of  mentioning  every  source  of  error  which 
has  been  overcome  is  obvious,  for  the  presumption  in  case  of 
doubt  naturally  is  that  any  not  mentioned  have  been  over- 
looked. 

The  restrictions  of  space  or  time  often  compel  the  omission 
of  such  data  from  printed  articles  or  from  papers  to  be  read. 
In  such  case,  however,  nothing  can  excuse  the  omission  of  a 
definite  statement  of  the  estimated  accuracy.  Failure  to  give 
the  complete  data  can  be  ascribed  only  to  urgent  necessity  for 
condensation,  or  to  ignorance  or  neglect  on  the  part  of  the 
observer,  and  either  of  the  latter  two  casts  grave  doubt  on  the 
quality  of  the  work. 

That  an  estimate  of  the  accuracy  of  his  work  should  be 
carefully  made  by  the  observer  and  presented  along  with  the 
result  is  but  little  less  in  importance  than  that  the  measure- 
ment should  be  made.  The  small  proportionate  amount  of 
labor  thus  bestowed  is  far  more  effectively  expended  than  an 
equal  amount  upon  the  work  of  observing. 

Planning  of  Direct  Measurement. — In  laying  out  in  ad- 
vance any  direct  measurement  this  will  have  to  be  done  under 
one  or  the  other  of  the  first  two  of  the  foregoing  specifica- 
tions, viz.  to  obtain  the  result  by  a  specified  method  as  ac- 
curately as  is  practicable,  or  to  select  a  method  and  obtain  a. 


SOLUTION  OF  ILLUSTRATIVE  PROBLEMS.  37 

result  with  a  specified  accuracy  (which  may  again  be  of  course 
merely  the  best  practicable). 

The  method  of  procedure  is  clearly  suggested  by  the  pre- 
ceding discussions,  but  may  be  summarized  as  follows : 

(a)  Obtain  a  general  idea  of  the  proposed  method,  appa- 
ratus, and  conditions  of  work,  or  of  several  methods,  etc.,  from 
which  selection  may  be  made. 

(b)  Make  a  thorough   study  of  all  discoverable  sources  of 
error,  taking  preliminary  observations  if  necessary. 

(c)  Plan  the  sufficient  removal  of  all  determinate  sources 
of  error  or  determine  corrections  for  them. 

(d)  Take  the  final  series  of  observations. 

Proper  attention  to  these  preliminaries  is  the  sine  qud  non 
of  good  results.     The  work  involved  in  them  often  far  exceeds 

o 

that  of  taking  the  final  observations,  and  is  sometimes  discour- 
aging to  the  experienced  observer  as  well  as  to  the  beginner. 
Its  essential  character  must  however  not  be  overlooked. 
Without  it,  much  time  and  labor  is  inevitably  wasted  and  a 
result  of  inferior  accuracy — often  of  no  value  whatever — is  the 
outcome. 


SOLUTION    OF    ILLUSTRATIVE    PROBLEMS. 
DIRECT    MEASUREMENTS. 

Example  III. — Problem.  The  weight  (or  mass)  of  an 
object  is  to  be  measured  by  an  equal-arm  balance  which  reads 
to  o.i  mgr.  The  result  is  desired  with  the  greatest  accuracy 
practicable  (see  Forms  of  Problems,  p.  33)  with  the  given 
balance. 

Solution. — First  (see  page  29)  unless  sufficient  data  are  in 
hand,  a  preliminary  series  of  weighings  is  made  to  find  the 
approximate  weight  W,  and  the  a.d.  of  the  single  weighings. 
Suppose  these  are  found  to  be  W=  34  grms.  and  a.d. 
=  o.oo  02 1  grms. 

Next   a   study  of   the    method   and   apparatus  would   be 


38  DIRECT  MEASUREMENTS. 

made.     Suppose  that  this  shows  that  the  work  is  subject  to 
the  following  determinate  sources  of  error : 

(1)  The  balance  arms  may  be  unequal. 

(2)  The  weights  may  not  be  standard  from  being  irregular 
among  themselves  and  by  the  unit  not  being  correct. 

(3)  The    temperature    may   be    unequal   throughout    the 
balance  case,   causing  inequality  of    balance    arms,   and    pro- 
ducing air  currents.     This  trouble  may  be  due  to  the  presence 
of  the  observer,  to  the  proximity  of  other  hot  or  cold  objects, 
to  the  attempt  to  weigh  a  body  warmer  or  cooler  than  the  air 
in  the  case,  or  to  other  causes. 

(4)  The  buoyancy  of  the  air  may  be  unequal  on  the  object 
and  weights,  owing  to  the  two  having  unequal  volumes. 

Of  these  (i)  may  be  removed  by  readjusting  the  distance 
apart  of  the  knife-edges,  or  eliminated  by  measuring  the  ratio 
of  the  arms  and  making  a  correction  for  it.  (2)  May  be  re- 
moved by  readjustment  of  the  weights,  or  may  be  eliminated 
by  comparing  with  standard  weights  and  applying  the  cor- 
rections thus  determined.  In  both  (i)  and  (2)  if  the  method 
of  removal  by  adjustment  is  adopted,  we  must  evidently  de- 
termine whether  the  readjustment  has  been  made  with  suffi- 
cient accuracy  by  measuring  the  ratio  and  by  testing  against 
standards  respectively.  (3)  The  disturbances  from  unequal 
temperature  may  be  reduced  by  observing  from  a  distance  if 
necessary,  and  by  securing  the  balance  from  other  sources  of 
radiation.  It  is  difficult  to  determine  when  the  disturbance 
from  this  source  is  sufficiently  removed.  (4)  The  buoyancy 
may  be  allowed  for  by  calculating  the  difference  of  weight  of 
air  displaced  by  the  weights  and  object  in  the  usual  manner 
This  will  require  an  approximate  knowledge  of  the  specific 
gravity  or  of  the  volume  of  the  object,  and  measurements  .of 
the  temperature,  pressure,  and  possibly  humidity  also,  of  the 
air  in  the  balance  case  at  the  time  of  weighing.  The  object 
must  of  course  not  be  losing  weight  by  evaporation  or  other- 
wise, or  gaining  by  condensation.  On  no  account  must  the 
object  be  at  a  temperature  considerably  different  from  that  of 


SOLUTION  OF  ILLUSTRATIVE  PROBLEMS.  39 

the  air  in  the  balance  case  or  the  latter  be  materially  different 
from  that  of  the  air  of  the  room. 

These  four  sources  of  error  must  be  reduced  to  be  of  negli- 
gible effect  compared  with  the  deviation  measure  of  the  result 
as  already  stated.  By  the  assumption  of  n  =  25  as  given  at 
page  29,  and  from  the  value  a.d.  —  0.00021  found  in  the  pre- 
liminary trial,  we  find  A.D.  =  - — :±r—  =0.000042.  This  i- 

1/25 

'  probably  beyond  the  limit  actually  attainable  on  the  balance 
reading  to  only  o.  I  mgr.  direct,  unless  the  method  of  weighing 
by  swings  is  used.  Moreover,  ordinarily  the  time  occupied  in 
making  25  independent  weighings  would  be  prohibitive.  It 
would  probably  give  more  nearly  the  practicable  limit  to  use 
n  =  4  to  9,  i.e.  A.D.  =  o.oo  oio  to  0.000070.  We  will  then 
use  A.D.  =  0.00007  grms.  as  an  approximate  value  of  the  d.m. 
To  be  negligible  then  the  residual  from  each  of  the  four 
sources  of  error  must  be  equal  to  or  less  than  %A.D./  Vp 
where/  =  4.  The  limit  is  therefore  J  X  o.oo  007/2  =  o.oo  ooi  2 
or  nearly  enough  o.oo  ooi  o  grms.  —  o.oi  mgr. 

(1)  Therefore  to  make  the  error   from  the  ratio    of   the 
balance  arms  negligible  this  ratio  must  be  adjusted  to  a  corre- 
sponding amount,  or  the  ratio  must  be  measured  for  a  correc- 
tion with  that  degree  of  accuracy.     The  ratio  of  the  balance 
arms  enters  as  a  direct  factor  and  is  very  nearly  unity.     An 
error  of  o.oo  ooi  grms.  in  34  would  be  0.0000003.     The  ratio 
must  then  be  determined  to  this  fraction,  i.e.  to  3  in  10000000. 
It  would  probably  be  impracticable  to  adjust  the  arms  as  close 
as  this  limit.     But  an  inspection  of  the  method  and  formula 
for  measuring  the  ratio  shows  that  this  limit  could  perhaps  be 
reached  in  that  measurement  by  making  several  observations. 

(2)  The  limit  of  o.oi  mgr.  in  the  adjustment  of  the  weights, 
or  in  the  comparison   of  them   with   a  standard,  cannot  be 
reached  with  any  means  ordinarily  at  hand,  if   at   all.     The 
best  that  can  be  done  without  undue  labor  is  probably  to  get 
the  errors  of  the  weights  within  about  o.i  mgr.     This  would 
be  a  residual  of  about  the  same  magnitude  as  the  d.m.  of  the 
weighing. 


40  DIRECT  MEASUREMENTS. 

(3)  The  disturbances  due  to  unequal  temperature  and  air 
currents  are  not;  easily  estimated.     They  would  be  rendered  as 
small  as  practicable  by  using  the  balance  in  a  room  of  nearly 
constant  temperature,  and  by  screening  the  balance  from  any 
objects  having  high  or  low  temperatures  and  from  draughts  of 
air.     It  would  be  advantageous   to  have   the  final    readings 
taken  by  the  observer  at  a  distance,  using  a  telescope  and  the 
method  of  swings.     The  amount  of  the  residual  could  prob- 
ably not  be  determined.     It  is  doubtful  whether  it  could  be 
reduced   to  the  limit  o.oi.     But   as  the  disturbances  would 
probably  not  be  of  the  same  sign  and  amount  at  different 
times,  they  would  appear  as  a  part  of  the  final  d.m.     The 
weighing  should,  of  course,  be  taken  on  different  days  and  at 
different  hours  in  the  same  day.     We  will  assume  this  residual 
to  be  negligible. 

(4)  For  the  buoyancy  correction  we  will  suppose  the  air  in 
the  balance  case  to  be  dry.     Let  the  specific  gravity  of  the 
substance  be  about  2,  that  of  the  weights  being  8.5.    Then  the 
volume  of  air  displaced  by  the  substance  would  be  34/2  =  17 
cc.,  that  by  the  weights  34/8.5  =  4  cc.     The  substance  would 
therefore  appear  too  light  by  the  weight  of  17  —  4  =  13  cc.  of 
air  at  the  density  of  that  in  the  balance  case  at  the  time.   Sup- 
pose the  observed  temperature  and   pressure  of  the  air  to  be 
16°  C.  and  760  mm.     The  weight  of  I  cc.  of  dry  air  under 
these  conditions  is    1.2  mgr.,  and  that    of   13  cc.  is  16   mgr. 
This  must  be  known  to  the  limit  o.oi  mgr.  in  order  that  the 
residual  be  negligible.     The  density  changes  by  0.004  mgr.  per 
cc,  for  i°  in  temperature,  so  that  for  13  cc.  the  change  would 
be  0.05  mgr.     The  limit  would  then  correspond  to  an  error  of 
\  =  o°.2  C.     The  change  of  density  of  air  per  mm.  change  of 
pressure  at  760  mm.  is  0.0015  mgr.  per  cc.     This  for  13  cc.  is 
O.O2  mgr.,  or  twice  the  limit.     As  both  temperature  and  press- 
ure are  variable  together  the  limit  must  be  o.oi/  1/2  ;  so  that 
the  accuracy  necessary  would  be  respectively  O.2/  1/2  =  O°.I4 
and  0.5  mm.  / 1/2  =  0.35  mm.     These  limits  could  be  reached 
only  with  great  care.     The  question  would  then  remain  as  to 
whether  the  weight  of  a  cc.  of  dry  air  was  known  with  suf- 


SOLUTION  OF  ILLUSTRATIVE  PROBLEMS.  41 

ncient  accuracy.  The  limit  required  is  o.oi  mgr.  in  0.016 
grms.,  or  about  0.05  per  cent.  It  is  doubtful  if  the  values 
for  the  density  of  air  given  in  the  tables  can  be  relied  on  as 
applying  to  ordinary  air  with  this  closeness,  especially  when 
the  humidity  of  the  air  is  neglected. 

It  appears  then  that  the  residuals  from  (i)  and  (4)  may  be 
rendered  nearly  but  not  quite  negligible,  that  the  residual  from 
(3)  cannot  be  well  determined,  but  is  assumed  to  be  negligible, 
and  so  far  as  it  exists  will  appear  as  a  part  of  the  d.m.,  and 
that  the  residual  of  (2)  will  be  about  equal  to  the  d.m.  Hence 
the  precision  measure  of  the  result  will  be  about,  but  prob- 
ably somewhat  greater  than 

6  =  |/(o.O7a  +  o.ioa)  =  o.i  2  mgr.  approx. 

There  will  be  no  constant  errors  large  with  respect  to  this,  so 
far  as  we  can  discover,  so  that  the  estimate  of  the  accuracy  of 
the  result  will  be  about  0.12  mgr. 

To  summarize  we  should  say  that  to  obtain  the  best  re- 
sult practicable  by  the  given  balance,  the  following  is  neces- 
sary : 

Ratio  of  arms  must  be  determined  to  6  in  10  ooo  ooo. 

Weights  should  be  corrected  to  at  least  o.i  mgr. 

Screening  from  radiation  and  draughts  should  be  thorough. 

Temperature  of  balance  case  must  be  measured  to  o°.i4  C., 
and  air  must  be  dry. 

Reduced  barometric  pressure  at  time  must  be  found  to  0.35 
mm. 

Constant  for  weight  of  air  must  be  known  well  within  o.i 
per  cent. 

The  result  will  then  be  accurate  to  about  o.  I  mgr. 

Example  IV. — Problem.  Desired  the  measurement  of  a 
voltage  x  of  about  1 10  volts  with  an  accuracy  of  o".2,  using 
a  Weston  magnetic  voltmeter  which  is  graduated  to  single 
volts  and  read  to  o".i  by  estimation,  the  conditions  being 
such  that  only  a  single  observation  can  be  taken  when  x  is 
being  read.  Resistance  of  voltmeter  about  17  ooo  ohms. 


42  DIRECT  MEASUREMENTS. 

Solution.  —  To  reach  this  limit  we  must  be  able  to  get  a 
p.m.  of  less  than  o*.2  when  all  determinate  errors  are  elimi- 
nated. 

We  must  first  find  the  deviation  measure.  Suppose  that 
several  measurements  made  on  a  constant  voltage  of  no". 
showed  an  a.d.  of  O".o6.  Lacking  this  test  we  should  probably 
assume  about  this  amount  as  the  a.d.,  for  the  following  reasons. 
The  deviation  would  be  due  to  errors  of  estimation  almost 
wholly.  If  the  index  were  fine  enough,  the  a.d.  of  estimating 
the  tenths  would  be  0^.025  (Special  Law  of  Deviations  I,  page 
21),  but  with  the  usual  size  of  index,  of  deviations,  and  the  un- 
avoidable parallax  0^.05  to  o*.i  would  be  a  safer  assumption. 

The  discoverable  sources  of  instrumental  error  may  be 
classified  as 

(1)  Changes  due  to  change  of  temperature  ; 

(2)  Permanent  alterations  of  resistance  ; 

(3)  Accidental  irregularities  in  spacing  the  graduations. 

(4)  Graduation  not  being  correct  volts,  whether  owing  to 
faulty  graduation  at  outset  or  to  change  of  strength  of  magnets 
and  springs. 

Of  these,  (i)  and  (3)  cannot  well  be  eliminated,  but  (2)  and 
(4)  can  be  determined,  (e.g.  by  comparison  with  a  Clark  cell) 
at  every  10  volts,  more  or  less,  along  the  scale  and  corrections 
applied.  The  points  whose  errors  are  thus  found  will  be  called 
the  calibrated  points,  and  the  corrections,  the  calibration  cor- 
rections. 

The  precision  measure  <?  being  of.  2  we  have 


and  to  put  in  the  work  to  the  best  advantage  we  shall  make 
r*  =  r*  =  rf  =  /*4a  approx.     Thus 

0.2-  =  0.06-  +  rI-  +  ri'  +  r.'  +  r4i, 

o.2a  —  o.o6a      0.036 

•*•  rf  =  raa  =  r^  =  r42  =  -  -  =  —  -  *-  =  0^.009; 

4  4 

.-.  r  =  0^.09  approx., 
which  is  therefore  the  normal  limit  for  the  residuals. 


SOLUTION  OF  ILLUSTRATIVE  PROBLEMS.  43 

(1)  By  the  statement  of  the  makers,  the  temperature  error 
for  the  magnetic  voltmeter  is  o.oi   per  cent,  per  degree  centi- 
grade.     At   1  10"   the  limit  0^.09  is  0.09/110  =  0.08   per  cent 
approx.      This  corresponds  to  a  change  of  8°  C.  in  the  tem- 
perature of  the  whole  instrument  ;  for  this  correction  is  not  for 
the  heating  of  the  coils  by  the  current  alone,  but  is  understood 
to  refer  to  a  change  of  the  whole  instrument.     Apart  from  the 
effect  due  to  greater  heating  of  the  coil  than  of  the  remaining 
parts,  this  error  might  be  made  negligible   by  observing  the 
temperature  and  correcting.     This,  however,  is  not  practicable 
and  would  probably  be  of  doubtful  value. 

(2)  An  accidental  change  of  resistance  of  the  coils  would 
cause  an  error  proportioned  to  the  change.     An  error  of  the 
limiting  amount,  that  is  of  0.08  per  cent,  would  be  produced 
then  by  an  accidental  change  of  0.0008  X  17000=  14™.     A 
change  of  half  this  amount  can  be  easily  detected  by  measure- 
ments on  a  bridge  from  time  to  time,  and  therefore  this  source 
of  error  can  be  made  negligible. 

(3)  These  irregularities  must  not  exceed  an  average  of  0^.09 
or    practically   o.i     division    between    the    calibrated    points. 
Their  amount,  however,  is  practically  not  determinable,  and 
their  effect  can  be  removed  only  by  calibration  at  many  points. 

(4)  The   errors  from  this  source  must  be  corrected  by  cali- 
bration with  Clark  cell  or  otherwise,  and  with  a  residual  not 
exceeding  cP.OQ.     The  Clark  cell  method  is  an  indirect  measure- 
ment, and  would    therefore   be   discussed  separately   by  the 
methods  later  given,  and  will  merely  be  summarized  here.     The 
expression  for  the  voltage  at  the  terminals  of  the  voltmeter  by 
one  method  is 


where  E  =  E.M.F.  of  cell  at  15°  C.,  a  =  temp,  coeff.  =  0.00036 
for  Carhart-Clark  cell,  t  =  temp,  of  cell,  R  =  res.  of  voltmeter, 
r  =  res.  between  terminals  of  cell.  The  results  of  such  a  dis- 
cussion  show  that,  for  a  result  accuracy  of  ov.i  in  V,  the  resid 


44  DIRECT  MEASUREMENTS. 

uals  for  the  various  component  quantities  must  be  as  follows : 
For  E,  o'.ooo56  (just  attainable) ;  for  a,  0.00008  (easily  made 
negligible);  for/,  i°.o  (easily  reduced  to  o°.5,  and  made  neg- 
ligible) ;  for  /?,  0.04  per  cent  (attainable) ;  for  r,  0.04  per  cent 
(attainable).  As  indicated  by  the  comments  in  the  parenthe- 
ses, we  can  do  perhaps  a  little  better  than  o77. 1  in  V.  But  the 
calibration  requires  also  a  reading  of  the  voltmeter  when  the 
voltage  is  V  at  its  terminals,  and  the  a.d.  of  this  reading  is  ov.o6. 
Hence  several  readings  must  be  taken  at  each  point  to  be  cal- 
ibrated. If  this  is  done  we  may  expect  to  get  the  calibration 
error  with  a  residual  not  much  exceeding  o".i  or  o".  15. 

Taking  the  d.m.  and  all  the  residuals  into  consideration  we 
may  then  hope  to  get  an  accuracy  as  follows  : 

<?'  =  o.o6a  +  o.  ia  +  o.O  +  0.1*  +  o'.  1 5  =  0.046. 
.-.  d  =  o\2i ; 

that  is,  of  the  prescribed  amount.  Evidently  this  limit  cannot 
be  reached,  however,  without  great  care  and  the  best  instru- 
ments, and  it  is  not  to  be  assumed  without  experimental  proof 
that  the  instrument  will  remain  long  without  a  change  exceed- 
ing this  amount. 


INDIRECT    MEASUREMENTS. 


Estimate  of  Accuracy  of  Indirect  Result. — The  terms 
accuracy  and  precision  are  used  with  the  same  significance  in 
connection  with  indirect  as  with  direct  measurements. 

When  the  estimated  accuracy  of  an  indirect  result  is  stated 
to  be  of  a  certain  amount  it  is  thereby  meant  that  the  pre- 
cision measure  of  the  result  is  of  that  amount,  and  that,  so  far 
as  can  be  discovered,  there  is  no  constant  error  in  the  result 
which  is  sensible  (i.e.,  not  negligible)  compared  with  this  pre- 
cision measure. 

When  it  is  stated  that  a  result  is  desired  with  a  specified 
accuracy,  it  is  similarly  meant  that  the  precision  measure  must 
not  exceed  that  amount,  and  that  no  discoverable  constant 
error  of  an  amount  not  negligible  in  comparison  with  this  must 
be  left  in  the  result. 

Thus  by  stating  the  accuracy  we  do  not  mean  that  the 
actual  error  of  the  result  is  of  just  that  amount,  for  if  we  did 
we  should  correct  accordingly.  Neither  do  we  mean  that 
this  is  a  more  probable  value  of  the  error  than  any  other. 
But  using  the  average  deviation  as  the  deviation  measure, 
we  mean  that,  so  far  as  we  can  discover,  the  average  effect  of 
all  the  errors  remaining  is  of  the  stated  amount,  and  may  be 
either  positive  or  negative.  This  implies  that  if  several  results 
were  to  be  obtained  under  the  same  conditions  by  the  same 

45 


46  INDIRECT  MEASUREMENTS. 

method,  apparatus,  etc.,  the  average  discrepancy  amongst  them 
would  be  approximately  of  this  amount. 

Thus  to  be  able  to  estimate  the  accuracy  of  an  indirect 
result,  we  must  have  data  for  finding  properly  the  precision 
measures  of  its  component  direct  measurements,  and  we  must 
be  able  to  compute  the  precision  measure  ot  the  result  from  that 
of  the  components.  Also  we  must  be  able  to  show  that  due 
care  has  been  taken  in  the  correction  or  removal  of  all  the 
determinate  errors  of  the  components,  so  that  none  but  negli- 
gible  constant  errors  remain.  Finally,  we  must  be  able  to 
show  that  the  "  error  of  method  "  (see  next  paragraph)  is  neg- 
ligible, or  if  not  so  to  take  it  into  account. 

The  numerical  part  of  the  estimate  of  accuracy  will  be 
the  P.M.  if  the  "  error  of  method  "  is  negligible,  or  will  be  the 
y(P.M*  +  R*)  if  R  be  the  estimated  amount  of  this  error. 

Error  of  Method. — Besides  the  constant  error  of  the  com- 
ponents there  is,  in  certain  cases,  another  source  of  error  in 
indirect  results.  The  method  adopted  may  have  inherent 
sources  of  error ;  that  is,  it  may  fail  to  give  correct  results, 
however  accurate  the  components,  because  the  result  sought 
does  not  bear  the  supposed  relation  to  the  components,  as 
expressed  by  the  function  from  which  the  result  is  computed. 
This  may  arise  through  the  introduction  of  some  approxima- 
tion, through  the  existence  of  some  inexact  hypothesis,  or 
from  other  similar  cause.  Errors  of  this  sort  will  be  called 
errors  of  method.  Their  amount  or  the  average  uncer- 
tainty arising  from  them  may  sometimes  be  estimated  and 
allowed  for.  The  possibility  of  their  existence  should  not 
be  overlooked.  As  an  instance,  we  may  cite  the  case  of  the 
"  Stray  Power  Method "  of  testing  dynamos  or  motors  (ex- 
plained among  the  final  illustrative  problems).  In  this 
method  the  assumption  is  made  that  the  "  stray  power  "  is 
the  same  when  the  machine  is  running  under  no  load  as  when 
under  full  or  partial  load,  certain  conditions  being  fulfilled. 
The  results  by  this  method  will  be  more  or  less  uncertain  if 
this  assumption  is  not  strictly  true. 


PRECISION  MEASURE   OF  RESULT  AND   COMPONENTS.   47 

Check  methods  are  of  course  of  the  same  importance  in 
detecting  and  eliminating  constant  errors  of  indirect  as  of 
direct  measurements.  To  obtain  the  most  accurate  results 
and  to  get  clues  to  constant  errors  independent  results  should 
be  obtained  by  as  many  different  methods  as  possible.  Such 
checks  may  be  had  by  changing  the  methods  or  instruments 
for  measuring  the  components,  or  by  changing  the  indirect 
process  for  another.  As  an  instance  of  the  latter  way  we  may 
take  the  determination  of  the  commercial  efficiency  of  an 
electric  motor.  Several  checks  on  this  might  be  obtained  by 
using  the  various  purely  electrical  methods,  and  still  other 
checks  by  testing  on  a  cradle  dynamometer,  with  a  friction 
brake,  etc, 

Relation  between  the  Precision  Measure  of  Result  and 
Components. —  Types  of  Problems.  The  following  three  types 
of  problems  are  the  chief  ones  which  arise  concerning  the  re- 
lation between  the  precision  measure,  P.M.,  of  the  result  and 
those,  p.m.,  of  its  components. 

First.  To  find  the  P.M.  of  the  result  given  the  p.m.  of  each 
component. 

Second.  To  find  the  best  value  for  the  p.m.  of  each  component, 
for  a  prescribed  P.M.  of  the  result. 

Third.  Given  the  P.M.  of  the  result,  or  the  p.m.  of  some  of 
the  components,  or  both,  to  find  the  best  magnitudes  ml ,  m^ ,  etc., 
of  the  components. 

The  third  of  these,  when  solvable,  enables  us  to  determine 
in  advance  the  best  relative  or  actual  magnitudes  of  the  com- 
ponents, that  is  those  which  will  yield  the  most  precise  result 
with  the  given  apparatus.  Examples  of  this  occur  in  such 
cases  as  finding  the  best  deflection  to  employ  in  using  an 
ordinary  tangent  galvanometer,  the  best  deflections  to  use  in 
rinding  battery  E.M.F.  or  resistance  by  the  two-deflection 
method,  etc.  These  problems  will  be  again  taken  up  after  the 
first  two  types  have  been  entirely  discussed. 

The  next  step  will  therefore  be  to  establish  formulae  for 
the  relation  between  the  P.M.  of  the  result  and  the  p.m.  of  the 
components  for  the  solution  of  the  first  two  types  of  problems. 


48  INDIRECT  MEASUREMENTS. 

General  Formulce.  Let  x,  y,  2,  .  .  .  represent  a  number  of 
independent  variables.  Then  if  w  is  a  quantity  which  is  some 
function  f  of  these,  this  fact  is  ordinarily  indicated  by  an  ex- 
pression of  the  form 


This  expression,  and  others  deducible  from  it,  apply  immedi- 
ately to  indirect  measurements  and  their  precision  discussion. 
For  example,  if  g  were  determined  by  means  of  a  simple 
pendulum,  the  process  would  be  an  indirect  measurement  of  g. 
The  component  quantities  directly  measured  would  be  /,  the 
length,  and  /,  the  time  of  a  single  vibration  of  the  pendulum  ; 
and  the  function  by  which  g  is  deduced  from  these  would  be 


,. 

f 


This  corresponds  to  the  above  general  expression  thus :  g  to  wy 
I  to  x,  and  /  to  y  ;  for  /  and  t  are  independent  quantities,  and 
it  is  a  constant. 

Instead  of  the  ordinary  mathematical  notation  using  x,  y, 
and  z  as  in  equation  (16),  it  will  be  much  more  convenient  for 
this  work  to  employ  a  special  one  adapted  to  the  purpose. 
We  will  then  use  as  the  general  expression 

M  =  f(ml9  m2, . . .  ,wn), [17} 

where  M  is  any  quantity  whatever  which  is  a  function  of  the 
quantities  *», ,  m^  ,...,;#„,  each  of  which  is  directly  meas- 
ured and  is  wholly  independent  of  the  others. 

A  quantity  M  thus  determined  will  here  be  called  an  indi- 
rect or  an  indirectly  measured  quantity.  It  is  often  called  a 
derived  quantity. 


SEPARATE  EFFECTS.  49 

Notation.  —  The  following  notation  will  be  always  em- 
ployed :  — 

M  =  the  final  indirect  result. 
m  =  any  component. 
a,  b  =  constants. 

n  =  the  number  of  directly  measured  components. 
A  =  any  small  finite  change  (expressed  in  units)  in  M. 

<J   ==       "  "  «  «  «  U  «  «      ^ 

A 

jp  =  the  corresponding  fractional  change  in  the  result  -/I/. 

-  =    "  "  "  "       "  any  m. 

m 

AM  and  dm  will  be  used  instead  of  A  and  d  when  needed 
for  greater  clearness. 

Any  change  will  be  considered  small  which  does  not  ex- 
ceed a  few  per  cent  of  the  corresponding  quantity. 

/()  will  be  written  for  brevity  in  place  of  /(m^m^,  .  .  .  mn). 

Corresponding  subscripts  will  denote  corresponding  quan- 
tities. Thus  A  i  will  denote  a  change  in  M  corresponding  to  a 
change  (^  in  ml  ,  etc.,  and  vice  versa. 

The  subscript  k  will  be  used  to  denote  a  general  term. 
Thus  mk  will  denote  any  component,  and  8k  or  Ak  the  corre- 
sponding changes  in  M. 

Separate  Effects.  —  I.  What  will  be  the  change  Al  in  M 
corresponding  to,  or  produced  by,  a  change  6^  in  ml  ,  all  other 
components  remaining  unchanged  ? 

Differentiating  f(  )  with  respect  to  ml  ,  all  other  compo- 
nents being  considered  constant,  we  have  as  the  rate  of  change 
of  M  with  ml 

dM  _df() 
dm    ~ 


Passing  from  the  limit  to  finite  changes  J,  and  tf,  ,  we  have 
A        dM 


50  INDIRECT  MEASUREMENTS. 

the  approximation  being  closer  as  the  changes  Al  and  tf,  are 
smaller.     Hence 


, 

8,  "  dm, 


Similarly  for  a  change  #a  in  #z3  ,  the  corresponding  change  in 
M  will  be 


and  so  on  for  all  the  n  components. 

Example  XIII,  page  86. 

II.  Given  a  small  change  A  in  M,  what  change  $  in  any 
component  m  alone  would  correspond  to,  or  would  be  neces- 
sary to  produce,  this  change  A  ?  This  is  evidently  merely  the 
converse  of  I. 

Consider  first  m^.     By  1  8  we  have 

4  _<*/() 

*,  "  dm',' 


Similarly, 


and  so  on  for  all  the  n  components. 

The  changes  Al  ,  J2,  etc.,  may  or  may  not  be  equal. 

Example  XIV,  page  86. 

Resultant  Effects.  —  III.  Suppose  that  the  small  changes 
Al9  Att  etc.,  occur  simultaneously,  what  will  be  the  resulting 
change  in  Ml  Here  we  must  distinguish  two  cases. 


RESULTANT  EFFECTS.  $1 

1°.  Where  the  changes  Aiy  z/a ,  etc.,  are  of  specified  magni- 
tude and  sign. 

2°.  Where  we  wish  to  deduce  a  general  expression  which 
will  give  us  the  best  solution  when  all  that  we  know  respect- 
ing the  Al ,  J,,  etc.,  is  the  following.  Each  A  may  be  of  either 
sign,  -f~  °r  — >  and  of  a  magnitude  following  a  certain  law  of 
distribution,  i.e.  the  J's  follow  the  "general  law  of  devia- 
tions." It  is  this  second  case  which  gives  us  the  formulae  to 
be  used  in  precision  discussions. 

i°.  For  the  first  case  we  know  that,  as  mlf  mt,  etc.,  are  in- 
dependent, the  resultant  change  A  in  M  due  to  the  simul- 
taneous occurrence  of  the  changes  A^ ,  A^ ,  etc.,  will  be  the 
algebraic  sum  of  these  separate  changes,  as  may  be  denoted  by 

A  =  A^  +  A%+.  .  .  An [23} 

This  expression  is,  however,  almost  never  made  use  of  in  the 
following  work,  as  the  conditions  for  which  it  holds  rarely 
occur. 

2°.  For  the  second  case,  the  desired  expression  may  be 
reached  as  follows. 

If  the  changes  A^ ,  A^,  etc.,  going  to  make  up  A  follow  the 
general  law  of  deviations,  then  by  the  Method  of  Least  Squares 
it  is  shown  that  the  best  or  most  probable  value  of  A  will  be 
given  by 

A2  =  Ax2  +  A,3  +  •  •  .  +  A«2 [24} 

By  "  most  probable  value"  it  is  meant  that  the  value  would 
have  greater  probability  than  one  obtained  by  any  other 
method ;  or  in  other  words,  if  we  had  the  means  of  putting  it 
to  proof  we  should  find,  in  an  indefinitely  long  experience,  that 
the  value  of  A  given  by  the  formula  would  be  nearer  the  truth 
than  one  obtained  by  any  other  expression.  This  value  of  A 


52  INDIRECT  MEASUREMENTS. 

is  the  best  in  the  same  sense  that  the  arithmetical  mean  is  the 
best  representative  value  of  a  series  of  observations. 

The  quantity  A  yielded  by  [24]  is  necessarily  one  of  the 
same  kind  as  A^  ^,,  etc.,  that  is,  if  Jlf  A^  etc.,  are  quantities 
following  the  general  law  of  deviations,  or,  in  other  words,  if 
they  are  deviations,  as  they  must  be  in  order  that  [24]  should 
be  applicable,  then  A  is  a  resultant  deviation  in  M.  Thus 
various  values  of  A  corresponding  to  various  possible  values 
of  A11  ^,,  etc.,  would  follow  the  general  law  of  deviations. 
Thus  A  is  always  a  quantity  of  the  same  nature  as  J,  ,  A^  ,  etc. 

Furthermore,  J,  ,  J3,  etc.,  are  quantities  following  the  same 
law  of  distribution  as  #,  ,  <?3  ,  etc.  That  is,  if  tf,  is  a  deviation, 

Al  will  follow  the  law  of  deviations,  for  Al  =  —  ^  —  *£t  ,  where 
is  a  constant. 


Thus,  if  8l  ,  rfa  ,  etc.,  are  simultaneous  deviations  in  w,  ,  m^  , 
etc.,  -J,  by  [24],  is  the  resultant  corresponding  deviation  in  M. 

Substituting  therefore  for  Al  ,  etc.,  their  expressions  in 
terms  of  tf,  ,  etc.,  given  by  19  and  20,  we  have 


This  expression  then  gives  us  the  most  probable  value  of  the 
deviation  in  M  which  is  the  resultant  effect  of,  or  corresponds 
to,  the  simultaneous  deviations  #,,#,,  etc.,  in  the  components, 
all  of  these  deviations  following  the  general  law  of  distribution 
of  deviations. 

These  two  expressions,  24  and  25,  are  the  fundamental  ones 
which  underlie  all  the  following  work.  The  solution  given  by 
the  expression  24  is  not  an  exact  one.  The  method  and  con- 
ditions under  which  it  is  deduced  are  such  that  we  know  that 
it  gives  merely  the  most  probable  value  of  J,  that  is,  it  gives  in 
the  long-run  better  values  of  A  than  any  other  expression  gives 
for  the  same  conditions  respecting  the  d's.  We  know  further 
that  the  value  of  A  obtained  by  it  is  entitled  to  greater  weight 


EQUAL  EFFECTS.  53 

the  larger  the  number  n  of  the  components,  *»,,/«,,...  w«; 
the  more  closely  the  tf's  follow  any  one  of  the  laws  of  distribu- 
tion of  deviations  ;  and  the  smaller  the  value  of  A  as  compared 
with  M. 

Example  XV,  page  87. 

If  in  24  we  divide  both  sides  by  Mt  we  have 

=     *+*  +  '-'+*'    *   •   •    [26] 


an  expression  to  which  reference  will  occasionally  be  made. 

IV.  If  we  were  to  ask  for  the  resultant  effect  the  question 
analogous  to  II  for  separate  effects,  it  would  be,  what  simul- 
taneous changes  in  #,,#,,...,#*  would  produce  a  specified 
resultant  change  A  in  M?  It  is  obvious  that  we  might  have  an 
infinite  number  of  values  of  dlt  #2,  etc->  which  would  produce 
the  specified  A  for  any  given  function  f(  ).  It  is  therefore 
necessary  to  assign  some  further  condition. 

Equal  Effects.  —  For  reasons  which  will  appear  later,  we 
may  advantageously  restrict  our  inquiry  to  the  case  where  each 
6  produces  an  equal  effect  with  every  other  on  the  value  of  A, 
i.e.,  where  each  S  produces  in  M  the  same  change  as  every 
other.  This  will  evidently  be  the  case  when  in  24  we  have 


[27] 


as  A^  is  the  change  in  J/due  to  &19  A^  to  #2,  and  so  on. 

For  the  general  case,  then,  where  the  tf's  follow  the  law  of 
deviations  (corresponding  to  24  and  25)  we  have  the  following: 
For  a  specified  value  of  A,  the  values  of  #,,  #2,  .  .  .  ,  dn  which 
will  produce  this  A  under  the  condition  of  equal  effects  are 
such  that 

Ai  =  A2  =  .  .  .  =  A«  =  ±=  ,     ......    [28] 


where  n  is  the  number  of  components,  m,  entering  into  the 
given  function. 


54  INDIRECT  MEASUREMENTS. 

Substituting  the  expressions  for  $lt  #2,  etc.,  in  terms  of 
At,  etc.,  we  have  further 


df(} 


Example  XVI,  page  88. 

These  are  also  fundamental  expressions,  and  it  will  be  seen 
by  inspection  of  the  method  of  their  deduction  that  the  values 
of  d  arrived  at  by  them  form  merely  a  special  set  of  values  out 
of  the  infinite  number  of  possible  values  which  would  satisfy 
the  simple  condition  that  the  d's  must  follow  the  law  of  devia- 
tions. 

Applications  to  Precision  Discussions.  —  In  the  application 
of  these  formulas  to  indirect  measurements,  we  shall  let  #,  ,  #a  , 
etc.,  represent  the  precision  measures  p.m.  of  the  components. 
Then  A,  being,  as  shown,  necessarily  a  quantity  of  the  same 
kind  as  tf,  will  be  the  precision  measure  P.M.  of  the  indirect 
result. 

That  this  application  of  the  formulas  is  justifiable  may  be 
shown  as  follows. 

The  only  condition  imposed  in  the  deduction  of  the  fun- 
damental expression  A*  =  A*  -\-  etc.,  is  that  the  quantities  Al  , 
A^  ,  etc.,  follow  the  general  law  of  deviations.  But  as  shown  at 
page  52,  zf,  ,  J3,  etc.,  will  follow  the  law  of  deviations  if  the 
values  of  tf,  ,  tfa,  etc.,  do  so.  Now  the  p.m.  which  is  used  as  <5 
is  a  quantity  of  the  same  nature  as  the  deviation  measure, 
being  calculated  by  combining  the  d.m.  with  the  residuals, 
which  are  also  deviation  measures.  And  the  deviation  meas- 
ure follows  the  law  of  deviations,  being  merely  special  value 
of  the  deviations.  Thus  the  p.m.  and  therefore  the  values  of 


FORMULAE  FOR  GENERAL  AND  SPECIAL  FUNCTIONS.        5  5 

6lt  #2,  etc.,  and  consequently  of  4lt  J2,  etc.,  follow  that  law. 
Therefore  the  general  expression  is  applicable  when  we  use 
precision  measures  for  the  values  of  d. 

The  same  form  of  statement  would  be  true  whatever  kind 
of  precision  measure  were  employed,  but  it  is  obviously 
essential  that  in  a  given  case  the  6's  must  all  be  of  the  same 
kind.  For  instance,  if  the  average  deviation  is  employed  for 
one  component,  it  must  be  for  all  in  the  same  computation. 
But  we  must  note  on  the  other  hand  that  if,  for  example,  ml 
in  a  given  measurement  is  a  quantity  observed  but  once,  while 
mz  is  a  mean  of  several  observations,  the  deviation  measure  dl 
of  m1  must  of  course  be  deduced  from  the  average  deviation 
(a.d.)  of  a  single  observation,  and  the  precision  measure  #2  of 
;;/„  must  be  deduced  from  the  A.D.,  the  average  deviation  of 
the  mean.  For  the  a.d.  is  a  deviation  measure  of  the  same 
kind  and  order  relatively  to  the  single  observation  used  as  ml , 
as  the  A.D.  is  of  the  mean  result  employed  for  m^ .  The  result- 
ing value  of  M  must  be  regarded  as  a  single  observation,  and 
its  A  must  be  considered  as  a  a.d.  if  any  one  or  more  of  the 
components  is  a  single  measurement.  But  if  all  of  the  com- 
ponents  used  are  mean  results,  the  M  would  be  considered  a 
mean,  and  the  A  a  A.D. 

Formulae  for  General  and  Special  Functions  f(  ). — 
There  are  several  special  forms  of  the  f unction  f(  )  for  which 
it  is  advantageous  to  have  formulae  giving  A  in  terms  of  6,  or 
the  converse,  and  formulae  for  equal  effects.  For  complete- 
ness the  general  expressions  are  first  given  and  then  those  for 
the  special  functions. 


M  -  f(mi ,  m* , . . . ,  mn) [32] 

Separate  Effects, — 


$6  INDIRECT  MEASUREMENTS. 

Conversely 


Resultant  Effect,— 

A"  =  A?  +  4,'  +  .  .  .  +  4.'.  '  .     .    •     .     [35] 


Equal  Effects,  — 


=  J,  =  .  .  .  =  4,  =-     .....     [37] 

Vn 


^_   _  ^  4,  _     i      J 

~~-~-  ~~-' 


Examples  XIII-XVI  illustrate  the  application  of  these  form- 
ulae to  precision  discussions  if  the  term  precision  or  precision 
measure  be  substituted  for  "  change." 

M  =  ±  mi  ±  w2  ±  .  .  .  ±  mn.      ...    .........    [39] 

Separate  Effects,  — 


Conversely 

^=±4,         ^ 
Resultant  Effect,  — 


Equal  Effects,  — 


=  tfa  =  .  .  .  =  6M  =     -  .....     [43] 
Vn 


FORMULAE  FOR  GENERAL  AND  SPECIAL  FUNCTIONS.      $? 

Example  XVII,  page  88. 

For  separate  effects  formula  [40]  may  be  stated  in  words  as 
follows.  For  a  sum  or  difference  the  change  in  the  result  is 
numerically  equal  to  the  change  in  the  component,  and  of  the 

same  sign. 

Deduction.  -3—-  =  ±  tf , ,  etc.  Whence  by  substitution  in 
the  general  formulae  we  have  the  above  results. 


M  =  ami  -\-  bnii  +  .  .  .  -4-  km*  .............    [44] 

a,  b,  .  .  .  ,  k  —  constants,  and  may  be  either  +  or  —  or  of 
indeterminate  sign  and  of  any  magnitude.  This  case  therefore 
includes  the  preceding. 

Separate  Effects,  — 

4  =  <*<*,,     4  =  J*,,     ...,     An  =  kdn.   .     .     [45] 
Conversely 

^  =  ±4,     <y,  =  £jf,     .  .  .  ,     *.  =  -kAn.  .     .     [46] 

Resultant  Effect  — 

^  =  (aW  +  (l>W+...+(MHy  .....     [47] 

Equal  Effects,  — 

A 

ad,  =  bd*  =  .  .  .  =  kdn  =  —  i  .....     [48] 

Vn 


Deduction.     --L    =  a,  etc.,  and  substituting   these  in  the 
general  formula  gives  the  above  results. 


$8  INDIRECT  MEASUREMENTS. 

M  =a-m,-w2-  .  .  .  -m« [49] 

a  =  constant  factor. 
Separate  Effects, — 


Conversely 
Resultant  Effect, — 


Equal  Effects,  — 

(^         ^  <^          i     ^ 

r  Jj  o-i 


m^       m^  mn 

Example  XVIII,  page  89. 

For  separate  effects  the  result  may  be  put  into  words  by 
saying  that  for  any  factor  the  fractional  change  in  the  result 
is  equal  to  the  fractional  change  in  the  factor. 

Deduction.     -=  —  -  =  a-m^  .  .  .-mn  =  —  ,  etc.,  and  substitut- 


ing  in  33  gives  4  ==•*,,.-.  ^=^.     Similarly,  jj  =  ^, 

etc.,  for  separate  effects.     Substituting  these  in  the  general 
formulae  36  and  38  gives  52  and  53. 


Separate  Effects, — 
A       <£ 


M~m;     M~m,' [53} 

4.=  -?!       4-=- 

M  ~        m^     M  m^ 


, [56] 


FORMULA  FOR  GENERAL  AND  SPECIAL  FUNCTIONS.       59 
Converse  is  evident. 

Resultant  Effect  — 

Same  as  52  since  negative  signs  disappear  on  squaring.  [57] 

Equal  Effects,  — 


ml  m^      m^  m^ 

N.B.  —  The  signs  are  of  no  importance  and  may  be  neglected 
in  most  precision  discussions,  for  they  merely  indicate  whether 
a  -(-  change  in  the  result  is  caused  by  a  +  or  a  —  change  in  the 
component,  a  fact  usually  of  no  interest. 

Deduction. 


dm 


-        .IS  before  • 

—        ,  da  uciurc  , 


and  so  on.     Substituting  in  general  formulae  gives  above  re- 
sults as  beforeo 


=  a-mv  .....    .....    .    .....        ....    159] 

a  ==  constant  factor  ;  v  =  constant  exponent,, 
Separate  Effect,— 


In  words:  If  the  function  be  a  constant  power,  and  be 
either  with  or  without  a  constant  coefficient,  the  fractional 
change  in  the  result  is  equal  to  the  fractional  change  in  the 
component  multiplied  by  the  exponent  of  the  power0 

Conversely 

£  .Jf.£ 

m       v  M' 
Example  XIX,  page  90. 


60  INDIRECT  MEASUREMENTS. 

N.B.  —  As  the  value  of  v  is  unrestricted,  this  holds  for  a 
negative  exponent.     Thus  if  v  =  —  c, 


=am-<  =      ,    and          -  -•  •    • 


Deduction. 


df(}  M  M  ^  A  $ 

,      —  avm°~l  =  v—      .-.  A  =  v  —  tf  and  ^r  =  v~  . 

dm  m  m  M         m 


y  =  ti-'tn\v  •  .  .  .  •  tnnw' L^^J 

a  =  constant  factor ;  v,  w  =  constant  exponents. 
Separate  Effects, — 


'  *  *  *  *       Af  —  ** '  m  *       *     *     * 

Conversely 

d        i   ^  tfL         i  ^« 

i  _^  *  *  ___  _j  r/?^ti 

Resultant  Effect, — 


Effects  — 

d  8H          i    A 

v—  =  .  .  .  =  w  —  =  --=  -r-  ......     [671 


Example  XX,  page  91. 

A7".^.  —  As  v,  wt  etc.,  are  unrestricted,  any  of  the  exponents 
may  be  negative,  so  that  this  covers  the  case  where  any  of  the 
factors  are  in  the  denominator.  The  only  difference  will  be 
that  in  separate  or  equal  effects  the  sign  of  the  exponent  will 
become  negative,  as  shown  in  formula  [62]  above.  But  as 


FORMULA  FOR  GENERAL  AND  SPECIAL  FUNCTIONS.      6l 

the  sign  is  usually  of  no  interest  as  already  explained,  we  may 
omit  the  consideration  of  it,  and  in  using  this  formula  for 
factors  we  may  treat  a  factor  in  the  denominator,  i.e.,  with  a 
negative  exponent,  precisely  as  if  it  were  in  the  numerator, 
i.e.,  had  a  positive  exponent. 

This  case  evidently  includes  as  special  cases  all  the  foregoing 
functions  separable  into  factors. 

Demonstration.  —  Obvious  from  the  two  preceding. 

Besides  the  foregoing  simple  functions  which  consist  merely 
of  the  sum  or  difference,  product,  quotient  or  power  of  the 
direct  quantities,  there  are  others  less  simple,  for  which 
formulae  for  equal  effects  are  of  much  service.  Of  these  the 
three  principal  ones  are,  first,  where  f(  )  can  be  separated  into 
a  series  of  factors  each  of  which  is  a  function  of  one  compo- 
nent direct  measurement  and  only  one  ;  second,  where  /()  can 
be  separated  into  two  or  more  terms  to  be  added  or  subtracted, 
each  of  which  is  a  function  of  several  of  the  components  but 
has  no  components  common  to  any  two  of  the  functions  ;  and 
third,  where  f(  )  can  be  separated  into  two  or  more  factors 
each  of  which  is  a  function  of  several  of  the  components  and 
having  no  component  common  to  any  two  of  the  functions. 
In  the  second  and  third  case  each  function  comprises  a  group 
of  components,  and  the  process  will  hence  be  referred  to  as 
separation  into  groups.  Of  course  any  of  these  functions  /() 
could  be  discussed  by  the  general  formulae,  the  advantage 
derived  from  the  use  of  the  special  ones  being  that  they  greatly 
lessen  the  work  of  differentiation  and  numerical  substitution, 
where  /(  )  is  complicated,  as  is  often  the  case  ;  and  they  also 
render  the  solution  of  the  problem  clearer  and  easier  to  follow. 

Separation  into  Factors  which  are  Functions  of  Single  Com- 
ponents. 
M  =  #m,)  .  p(ma)-.  .  .  -<r(m«)  .............    [68] 


Resultant  Effect  — 


i  A  v 
\>  '-'- 


M   --  '  '  '  '    [69] 


62  INDIRECT  MEASUREMENTS. 

Equal  Effects, — 

<y0(//2  )  ^P(^»)  ^0"(^Z«)  I        ^ 

>("O     =     pfaf.)     =  --'   =      a(m  )     =^J/'  '        t7°] 

f  \      i/  '  \      a/  \      »/  r  /* 

These  determine  the  values  of      .     !  ,  etc.,  from  the  stated 

value  of  -jy=  ,  and  from  these  values  we  have  to  find  the  corre- 
j\ 

spending  values  of  —  or  of  #,  as  the  case  may  require,  by  the 
m 

general  or  special  formulas  for  separate  effects. 
Example  XXI,  page  91. 


Deduction.— A  =-±+-d..    Now 

M 


Y*T  approx.  = 
Similarly, 


M  '      PK)  '  e 

By  [26]  for  resultant  effect  we  must  have 


in  which  substitution  of  the  above  values  gives  [69]. 
By  [38]  for  equal  effects  we  must  have 

M  "~  M '  ~  ~  M  ~~  fa  M  ' 

A      A 
from  which  by  substituting  the  foregoing  values  of  j± ,  -j^,  etc., 

we  obtain  the  desired  expression  [70]. 


FORMULA  FOR  GENERAL  AND  SPECIAL  FUNCTIONS.      63 

Separation  into  Groups. 

.  .  .  ,  mp)  ±  p(mq,  .  .  .  ,  ms)  ±  .  ,  .  ± 


where  there  are  p  components  in  the  function  0(  ),  r  in  the 
function  p(  ),  s  in  the  function  cr(  ),  and  n  in  all. 
Resultant  Effect  — 


Equal  Effects,  — 


or 


These  serve  to  determine  the  values  of  ^40(  ),  4p(  ),  etc., 
from  the  stated  value  of  A.  And  from  them  we  should  pro- 
ceed to  find  the  corresponding  values  of  tf,  ,  tfa  ,  etc.,  by  the 
general  or  special  formulae  for  separate  effects. 

Deduction.  —  Adhering  to  the  same  notation  as  elsewhere 


In  the  second  member  of  this  expression  the  first  group 
obviously  gives  [^0()]a,  the  second  [^p()]a,  the  last  [Jcr()]2, 
so  that  we  may  substitute  these  expressions,  obtaining  for- 
mula [72.] 

For  equal  effects  of  tf, ,  <9,  ,...,<?„  in  the  components,  we 
must  have 


64  INDIRECT  MEASUREMENTS. 

Hence 


and  similarly  for 


Hence  for  equal  effects  we  must  have 

Vr-~,    ...,  - 

Vn  Vn 

as  above  given  ;  or  transposing  each  term  and  equating, 


M  =  <|>(m,  ,  .  .  .,  »ip)      p(mg  ,  .  .  .  ,  m«)      .  .  .      <r(mt9  ...  9  mn), 


where  there  are  /  components  in  0(  ),  r  in  p(  ),...,  s  in  <r(  ). 
Resultant  Effects,— 


• 

Equal  Effects,— 


or 


_    />  ^.   4°()_    A  A        4fQ-    /i  ^ 
~\Jn'M'     f()-y  n    M"      '  <r(  )  ~  V  »  '  Jf 


These  determine  the  values  of       ,\  ,  etc.,  whence  the  cor- 


FORMULA  FOR  GENERAL  AND  SPECIAL  FUNCTIONS.        65 

^ 
responding  values  of   -  -  or  d,  as  the  case  may  require,  are 

computed  by  the  separate  effects  formulae. 
Example  XXII,  page  94. 
Deduction. — By  the  general  formula  [36] 


A 
To  find  the  values  of  -=-k  etc.,  we  have 


Dividing  both  sides  by  M  and  passing  from  the  limit  to  finite 
changes,  we  have  approximately 


00  ' 


where  rf,0(  )  denotes  the  change  in  <p(  )  corresponding  to  tf,  in 
m^  .     Similarly  for  the  other  terms  we  have 


- 

.  If  • 

Substituting  these  values  in  the  above  expression  we  have  tor 
the  first  group 


66  INDIRECT  MEASUREMENTS. 

rj0(ri* 

But  this  second  member  obviously  expresses  I    ^  \    I  •     And 
similarly  the  second  and  last  groups  yield 


p 

Hence  for  combined  effects 


which  was  to  be  proved. 

For  equal  effects  of  tf,  ,  tfa  ,  .  .  .  ,  dn  ,  we  must  have 


_          , 

^. 

~  ' 


n 


.         ,       ,         _ 
"        " 


* 


Hence  for  equal  effects  we  must  have 


_ 
n'M>      p()~-=\n'M'     '"    <r()~n 

or  transposing  and  equating, 


which  was  to  be  proved. 


CRITERIA    FOR  NEGLIGIBILITY.  6? 

Criteria  for  Negligibility  of  8  in  Components. — Any  small 
change  6k  in  any  single  component  mk  is  negligible,  i.e.  insig- 
nificant in  its  effect  on  the  accuracy  of  M,  when  the  corre- 
sponding change  Ak  in  the  result  M  is 

4k  <  £4 [79] 

Any  number  p  of   such  changes  tf, ,  #2 ,  etc.,  are  negligible 
when  the  change  corresponding  to  any  one  of  them  is 


or,  more  strictly,  when 


where  Aa  ,  JA  ,  etc.,  are  the  changes  in  M  corresponding  to  the 
changes  da  ,  db  ,  etc.,  in  the  components  ma,mb,  etc. 

The  changes  Jl  ,  ^a,  ^,  etc.,  in  J/  are,  of  course,  those 
which  enter  into  the  general  expression 


df(\ 
Thus,  in  general,  4  =  -^—  .  (^  ,  etc.,  or  these  may  have  in  spe- 

cial cases  the  special  values  given  in  formulae  [33]  to  [67]  ;  so 
that  we  may  find  dl  ,  etc.,  having  dl  ,  etc.,  given,  or  may  find  the 
value  of  tf,  corresponding  to  the  negligible  amounts  ^  ,  etc. 

As  we  can  use  precision  measures  for  tf's,  the  foregoing  state- 
ments  form  the  criterion  by  which  we  may  determine  when 
the  precision  measure  of  one  or  of  any  group  of  components  is 
negligible  in  its  effect  on  the  P.M.  of  the  result. 

We  have,  of  course,  as  a  special  case  of  the  above  general 

dfc 

one,  6k  or  —  negligible  when 
mk 


68  INDIRECT  MEASUREMENTS. 

and  for  p  such  changes  when  the  effect  corresponding  to  any 
one  of  them  is 


or,  more  strictly,  when 


Example  XXIII,  page  96. 

Deduction.—  For  the  reasons  stated  at  page  17,  any  single 
change  in  A  which  affects  it  by  -^A  or  less  is  negligible.  In 
the  expression,  then, 


what  would  be  the  value  of  any  single  change,  4k,  which  would 
reduce  A  to  A  —  ^A,  that  is  to  o.cpJ  ?  The  corresponding 
change  in  A*  would  be 


j«  -  (0.90^)'  =  A*  -  0.8  1  ^3  = 
.-.  4^  =  o.  19  Ja,  4*'  =  0.44^  ; 

/.  J*  —  ^,  approximately. 

The  limit  Ak  —  %  A  is  a  safer  one  and  preferable  for  ordinary 
work  where  n  is  small.  It  corresponds  to  a  change  of  ^^. 
This  limit  of  \A  is  employed  in  the  present  demonstration. 

If  instead  of  a  single  change  Ak  there  are  p  changes  whose 
joint  effect  on  A  is  to  be  considered,  then  this  joint  effect 
would  be  equal  to  a  single  change  of  the  magnitude 


;  _|_  Af  +  .  .  .  +  A?.     For  the  effect   of   these   on  Ja   is 
equivalent  to  that  of  a  single  change  Ak  such  that 


CRITERIA    FOR   NEGLIGIBILITY.  69 

This  furnishes  the  general  limit,  the  last  one  given  in  the  cri- 
terion.    A  special  case  of  this  is,  of  course,  when  Aa  =  Ab  =  .  .  . 


=  Ap  ,  so  that  VA*  +  A?  +  .  .  .  +  Af  =  ^pAa  —     pAb  =  etc. 
Whence  for  Ak  —  \A  we  have 


This  is  a  more  convenient  limit  to  apply  practically,  but 
obviously  it  is  not  essential  that  this  equality  should  exist,  so 
that  this  is  an  arbitrary  or  more  special  criterion.  For  reasons 
which  will  be  shown  in  the  section  on  equal  effects,  however, 
this  arbitrary  limit  will  not  be  widely  departed  from  even  under 
the  general  limit. 

Corresponding  criteria  can  be  given  for  negligible  compo- 
nents m.  Thus  cases  will  be  found  where  the  effect  of  one  of 
the  components  mk  itself  upon  the  result  M  is  so  small  that  the 
component  may  be  neglected  altogether  in  the  computation  of 
M  and  its  measurement  may  be  omitted.  This  would  evidently 
be  the  case  when  the  omission  of  mk  did  not  affect  M  by  an 
amount  exceeding  -^A,  or  better  ^J,  or  by  %A/Vpt  where  there 
are  p  components  to  be  simultaneously  omitted.  This  last 
criterion  applied  by  direct  inspection  is  the  best. 

A  convenient  preliminary  criterion  is  as  follows  :  Find  the 
value  of  Ak  which  would  be  negligible  under  the  criterion 
Ak  <  \A.  From  this  deduce  the  corresponding  value  of  6k. 
Then  if  mk^dk,  it  may  be  usually  wholly  neglected.  This 
should  therefore  be  checked  by  the  preceding  criterion,  since 
it  is  not  strictly  true  for  certain  functions. 

If  it  is  a  question  whether  several  components  may  be  neg- 
lected, then  if  /  is  the  number  of  these,  the  limit  for  dk  must 

obviously  be  that  corresponding  to  -  —  instead  of  \A. 

3  vp 

The  opportunity  for  such  omissions  occurs  most  frequently 
in  functions  containing  correction  terms,  such  as  temperature 
corrections  or  those  in  the  formula  for  the  tangent  galvanom- 
eter. 


70  INDIRECT  MEASUREMENTS. 

Numerical  Constants. — A  large  number  of  functions  con- 
tain numerical  constants,  either  mathematical  such  as  n,  or 
physical  such  as  the  density  of  a  given  substance,  etc.,  whose 
numerical  values  are  known  beyond  the  requirement  of  the 
case.  For  these  we  desire  to  know  how  many  places  of 
significant  figures  it  is  necessary  to  retain  in  the  computation. 

The  error  which  enters  by  them  is  only  that  due  to  the 
rejected  figures,  e.g.,  from  using  3.142  for  n  instead  of 
3. 14  1 59  265  ...  As  the  number  of  such  constants  in  any  given 
function  rarely  exceeds  two,  we  may  safely  proceed  as  follows. 
From  the  prescribed  or  computed  value  of  AM  calculate  what 
would  be  a  negligible  value  dc  or  dc/c  of  the  constant  just  as 
though  it  were  a  directly  measured  component.  Reject  all 
places  which  do  not  affect  the  constant  beyond  this  limit,  de- 
parting if  at  all  on  the  side  of  safety. 

Example  XXI 71,  page  96. 

Equal  Effects. — Demonstration.  In  deducing  formulae  [27] 
etc.,  the  condition  was  arbitrarily  imposed  that  we  should 

A 

have  A^  =/7 ,  =  .  .  .  =  Ak  =  .  .  .  =  An  =  — -,  in  which   case 

Vn 

the  d  of  each  component  has  an  equal  effect  on  AM.  In  what 
way,  and  to  what  extent,  this  is  the  best  guiding  condition 
when  d  represents  the  precision  measure  may  be  seen  as 
follows. 

If  the  number,  n,  of  components  were  indefinitely  large,  and 
if  the  tf's  followed  strictly  the  law  of  deviations,  the  best  values 
of  tfj  ,  #3 ,  etc.,  for  a  stated  value  of  A,  would  be  such  as  would 
render  the  total  labor  of  observing  a  minimum.  The  distribu- 
tion of  precision  would  thus  be  determined  by  the  relative 
difficulty  of  the  several  component  measurements,  and  would 
obviously  not  be  such  as  to  make  A^  =  A^=  etc.,  that  is,  it 
would  not  correspond  to  equal  effects.  The  values  of  the  J's 
of  the  more  difficult  measurements  would  be  allowed  to  be 
greater,  and  those  of  the  less  difficult  ones  would  be  made 
smaller  than  would  be  prescribed  by  that  condition.  Thus  the 
more  difficult  measurements  would  be  carried  out  with  a  lesser 


EQUAL  EFFECTS.  Jl 

precision  and  the  easier  ones  with  a  greater  precision  than 
would  correspond  to  equal  effects. 

It  is,  however,  not  practicable  to  frame  reliable  formulae 
which  shall  take  into  account  and  give  due  weight  to  the  dif- 
ficulty of  the  component  measurements.  If  such  formulas  were 
attempted,  they  would  be  based  upon  the  assumption  that  n 
should  be  large  and  the  law  of  deviations  strictly  followed, — 
conditions  which  would  not  be  fulfilled  by  the  usual  cases  in 
practice,  where  n  is  small  and  the  law  not  closely  adhered  to. 
Also  it;  would  be  necessary  to  assume  some  general  law  as  to 
the  increase  of  labor  with  increase  of  precision,  and  no  law 
would  apply  with  equal  exactness  to  all  kinds  of  measure- 
ments. 

Thus  although  in  any  specific  case  the  condition  of  equal 
effects  does  not  yield  an  exact  and  complete  solution  of  the 
best  values  of  the  ^Ts,  yet  it  will  give  the  best  solution  to  use 
for  the  point  of  departure  from  which  to  prescribe  the  pre- 
cision of  the  components.  The  tf's  thus  determined  should 
therefore  not  be  regarded  at  all  as  absolute  and  unalterable, 
but  merely  as  guiding  or  approximate  values  from  which  we 
should  depart  slightly  in  the  direction  of  less  precision  for  the 
more  difficult  and  greater  precision  for  the  less  difficult  com- 
ponents. Yet  it  is  also  clear  that  we  cannot  depart  widely 
from  this  condition.  For  if  we  increase  the  d  of  one  or  more 
components  we  must  diminish  that  of  some  or  all  of  the  others 
in  order  to  preserve  the  stated  value  of  AM,  and  the  limit  of 
this  increase  is  soon  reached.  Thus  to  take  an  extreme  case,  if 
Ak  be  the  effect  on  M  for  the  8  of  any  one  component  mk ,  and 
if  only  Ak  be  increased,  the  largest  value  which  this  can  have 
will  be  Ak  =  AM,  which  will  be  only  Vn  times  its  equal  effect 
value  AM/  Vn.  And  in  this  case  all  the  other  ^'s  must  be 
made  so  small  that  their  joint  effect  is  negligible.  Thus  if  n  = 
9,  which  is  greater  than  its  average  value,  it  would  be  possible 
in  the  extreme  case  to  increase  Ak  to  only  3  times  its  equal 
effect  value. 

This  extreme  limit  would  rarely  be  resorted  to,  first  because 
the  conditions  which  might  justify  it  seldom  arise,  second  be- 


72  INDIRECT  MEASUREMENTS. 

cause  of  an  objection  to  increasing  any  one  or  two  of  the  ^'s 
beyond  the  others,  a  reason  for  which  will  be  given  below. 

The  conditions  as  to  relative  difficulty  of  components  which 
might  justify  the  extreme  case  would  arise  only  when  the 
labor  saved  on  the  one  component  was  equal  to  the  added 
amount  on  the  others;  in  other  words,  when  as  much  labor  was 
required  to  determine  mk  with  a.  p.m.  producing  Ak  =  A  as  was 
necessary  to  reduce  the  total  resultant  effect  of  the  (n  —  i)  re- 
maining A's  to  a  negligible  amount,  viz.  \A  (formula  [80]).  Let 
Ae  denote  the  effect  of  the  p.m.  of  each  component  for  equal 
effects,  i.e.  A/  Vn.  Then  this  statement  means  that  if  all  the 
(n  —  i)  components  are  equally  difficult,  each  must  have  itsfl.m. 
correspond  to  \A/  Vn  —  I  or,  nearly  enough,  to  %A/Vn,  i.e.  to 
\Ae.  At  the  same  time  Ak  becomes  VnAe.  If  not  all  of  the 
(n  —  i)  components  are  equally  difficult,  the  latter  part  of  this 
statement  must  be  modified  accordingly. 

Ordinarily  there  would  be  unequal  difficulty  among  several 
of  the  components,  so  that  more  than  one  A  must  be  increased 
and  not  all  the  remaining  can  be  made  negligible  with  equal 
labor.  Hence  the  limit  of  Vnde  would  be  rarely  called  for. 
The  greater  the  inequality  the  narrower  the  allowable  limits  of 
change  of  the  ^'s  from  Ae.  A  change  of  A  to  2Ae  would  be 
unusual. 

There  is  an  objection  to  the  increase  of  any  one  or  two  of 
the  z/'s  to  the  extreme  limit  allowable.  For  when  any  A  is 
made  large  compared  with  most  or  all  of  the  others  the  distri- 
bution of  these  quantities  as  to  magnitude  ceases  to  conform 
at  all  approximately  to  the  law  of  deviations  and  the  applica- 
bility of  all  our  formulae  based  upon  that  law  is  thereby 
materially  reduced.  The  special  case  when  all  the  A's  are 
equal,  the  <Ts  being  precision  measures,  conforms  closely  to  that 
law,  owing  to  the  nature  of  the  precision  measure. 

Estimated  Precision  Measures  of  Components.  —  It  is 
sometimes  impracticable  to  make  more  than  a  single  measure- 
ment of  a  component  at  the  time  of  the  final  measurements, 
so  that  there  are  not  data  enough  for  finding  its  deviation 


COMPONENTS  WITH  SPECIAL  LA  WS  OF  DEVI  A  T10N.      73 

measure.  In  such  case,  it  is  desirable  to  take  beforehand  or 
afterward  a  series  of  observations  under  similar  conditions  to 
obtain  those  data.  For  this  purpose  it  is  seldom  essential  that 
the  value  of  m  should  be  rigidly  the  same  as  in  the  final  ob- 
servation, since  the  d.m.  usually  changes  but  little  with  small 
changes  in  m. 

It  is  often  impracticable  to  get  any  direct  data  whatever 
for  finding  the  d.m.,  but  frequently  when  this  occurs  an  esti- 
mate of  the  d.m.  can  be  made  in  some  way  which  will  serve 
fairly  well.  One  method  is  to  estimate  not  the  a.d.  but  the 
maximum  deviation  likely  to  occur  in  a  large  number  of 
deviations,  and  to  divide  this  maximum  by  4,  which  will  give 
roughly  the  a.d.  This  is  based  on  the  fact  that  the  average 
frequency  of  deviations  as  large  as  ^a.d.  is  only  I  in  1000,  ac- 
cording to  the  general  law  of  deviations. 

Components  with  Special  Laws  of  Deviation. — Compo- 
nents occasionally  are  met  with  whose  deviations  follow  the 
special  law  stated  at  page  24,  or  other  special  laws,  instead  of 
the  general  one.  The  formulae  for  the  relation  of  the  P.M.  of 
the  result  to  the  p.m.  of  the  components,  being  based  on  the 
general  law,  are  not  strictly  applicable  to  indirect  measurements 
containing  such  components.  Nevertheless  the  inaccuracy  in- 
troduced by  their  application  to  such  cases  will  be  hardly,  if  at 
all,  greater  than  is  very  often  introduced  by  the  failure  of  the 
actual  deviations  in  other  components  to  follow  the  general  law 
owing  to  the  smallness  of  the  number  of  the  observations,  or 
by  the  further  inaccuracy  due  to  the  usually  small  value  of  n. 
It  is,  therefore,  not  best  to  make  any  exception  of  these  special 
cases. 

f reparation  of  Functions  for  Discussion. — This  is  a  point 
of  the  utmost  importance.  Before  beginning  the  precision  dis- 
cussion of  any  indirect  measurement  the  function/(w, ,  . . . ,  mn) 
must  be  written  out  in  full.  It  must  be  put  into  such  form 
that  every  measured  component  appears  in  it,  and  the  form  must 
be  the  equivalent  of  that  used  for  computation.  If  the  latter 
is  done  by  steps,  then  all  of  these  must  be  combined  into  the 
general  formula.  If  any  of  the  measured  components  are  re- 


74  INDIRECT  MEASUREMENTS. 

lated  by  equations  of  condition,  these  equations  do  not  appear 
in  the  expression  for  f( ).  In  problems  other  than  those  on 
best  magnitudes  the  only  condition  as  to  independence  of  the 
components  about  which  we  are  concerned  is  that  they  shall 
be  obtained  by  independent  measurement. 

For  example,  if  we  were  determining  g  by  the  simple  pen- 
dulum we  should,  in  the  simplest  case,  observe  the  time  t,  of  v 
single  vibrations,  this,  number  being  counted,  and  we  should 
measure  the  distance  h  from  the  suspension  axis  to  the  top  of 
the  ball  and  the  vertical  diameter  d  of  the  ball.  The  first  form 
which  the  expression  to  discuss  should  receive  should  not  then 
be 


but 


since  /  is  not  directly  measured,  but  is  computed  by  means  of 
the  expression  in  the  parenthesis. 

As  however  v  would  be  a  whole  number  so  small  that  any 
miscount  would  be  of  the  nature  of  a  mistake  and  not  of  a 
deviation,  we  might  very  properly  not  regard  v  as  a  measured 
component,  and  if  we  chose  could  substitute  /,  the  time  of  a 

single  vibration,  for  -  . 


v 


The  measured  components  whose  #'s  would  be  discussed 
would  then  be  t  (or  /„),  h,  and  d\  not  /  and  /. 

After  the  complete  expression  has  thus  been  written  out  it 
would  receive  as  many  simplifications  as  possible  in  the  course 
of  the  discussion,  to  save  labor.  The  nature  of  these  changes 
will  be  further  explained. 

More  extended  illustration  of  this  matter  will  be  found  in 
some  of  the  later  examples. 


SIMPLIFICATION  OF  FUNCTIONS.  75 

Simplification  of  Functions. — Let  the  fact  be  recalled  to 
mind  that  the  solutions  of  all  problems  on  this  subject  are  to 
be  carried  out  to  only  two  places  of  figures  and  need  be  correct 
only  to  i  part  in  10,  and  that  the  results  are  often  (e.g.,  equal 
effects)  to  be  regarded  only  as  approximate  guides.  It  will 
then  be  seen  that  simplifications  of  the  original  functions  will 
be  allowable  to  save  labor,  and  that  they  may  be  of  a  much 
more  extreme  character  than  would  be  allowable  in  the  actual 
computation  of  the  results  of  the  measurement.  Moreover, 
different  simplifications  may  be  introduced  when  different  com- 
ponents are  being  discussed.  The  one  criterion  for  the  admis- 
sibility  of  any  proposed  simplification,  when  the  solution  is 
being  made  with  respect  to  any  component  mk,  is  that  the  value 
of  dmk  or  the  effect  of  Ak,  as  the  case  may  be,  must  not  be 
changed  by  more  than  i  part  in  10  by  the  simplification.  In- 
spection, or  a  preliminary  differentiation,  will  usually  show 
easily  whether  the  proposed  change  fulfils  this  condition. 

The  two  chief  methods  of  simplification  are  the  use  of  ap- 
proximate expressions  in  place  of  the  exact  form  for/(),  and 
the  omission  of  certain  terms  while  differentiating.  The  tables 
of  approximation  formulae  given  in  some  text-books  on  physi- 
cal manipulation  will  suggest  available  approximations.  Many 
other  ways  of  simplification  will  be  suggested  by  experience 
and  ingenuity  in  particular  cases. 

In  somewhat  complicated  functions  one  or  more  com- 
ponents, mk,  etc.,  may  occur  in  several  parts  of  the  expression, 
as  in  the  complete  formulae  for  the  tangent  galvanometer,  in 
the  expressions  for  the  efficiency  of  a  dynamo  by  any  of  the 
stray  power  methods,  etc. 

If  such  a  case  is  being  treated  by  the  general  method,  the 
differentiation  may  sometimes  be  simplified  either  by  omitting 
to  differentiate  with  respect  to  mk  those  terms  in  which  it  enters 
in  such  a  way  that  the  differential  will  be  negligible  (less  than 
o.  i  of)  compared  with  the  differential  of  other  terms  into  which 
mk  enters ;  or  by  omitting  the  differentials  of  such  terms  if  when 
made  they  are  found  to  be  thus  negligible.  Examples  of  this 
will  be  given. 


76  INDIRECT  MEASUREMENTS. 

If  the  case  is  treated  by  any  of  the  special  formulae,  terms 
in  which  mk  enters  in  this  secondary  way  may  be  neglected 
when  solving  with  respect  to  6mk.  But  it  must  be  borne  in 
mind  that  although  one  term  containing  mk  is  of  negligible 
(one-tenth)  magnitude  compared  with  another,  it  does  not 
necessarily  follow  that  the  effect  of  dmk  in  the  one  will  also  be 
negligible  compared  with  the  other.  Whether  this  will  be  so 
or  not  depends  upon  the  form  of  the  function  for  each  of  the 
two  terms.  Inspection  will,  however,  usually  show  plainly 
what  terms  are  negligible. 

In  separating  into  groups  (pages  61,62)  the  groups  need  not 
always  be  strictly  exclusive  with  respect  to  each  other.  That 
is,  a  given  component  mk  may  be  common  to  two  or  more 
groups  provided  that  it  occurs  only  in  a  secondary  way  in 
all  groups  but  one.  In  discussing  such  groups  mk  would  be 
studied  only  in  the  group  in  which  it  principally  occurred. 
Sometimes,  however,  a  wider  departure  than  this  can  be  made 
for  the  sake  of  simplifying  problems,  bearing  in  mind  that  the 
solutions  are  at  best  only  approximate  guides.  Thus  two  or 
more  groups  may  sometimes  be  allowed  to  contain  a  common 
component  and  an  allowance  made  for  the  fact  in  the  solution 
and  in  the  result. 

Example  XXIV. — See  also  Solutions  of  Illustrative  Prob- 
lems. 

Significant  Figures. — In  all  numerical  computations  it  is 
essential  to  know  how  many  significant  figures  to  retain  in  the 
data,  the  results,  and  at  each  step  in  the  computations.  Fail- 
ure to  follow  a  correct  and  consistent  practice  will  result  on  the 
one  hand  in  a  ludicrous  and  misleading  display  of  figures  and 
a  great  waste  of  time,  and  on  the  other,  in  an  ignorant  sacrifice 
of  the  accuracy  of  the  observations,  and  in  illusive  results. 
Simple  rules  can  be  deduced  which  avoid  either  extreme. 
Their  application  becomes  easy  and  almost  involuntary  after 
some  practice,  although  it  requires  close  attention  when  first 
undertaken.  The  following  deduction  of  these  rules  is  in  part 
a  special  application  of  the  foregoing  methods  and  is  given  as 
such. 


SIGNIFICANT  FIGURES.  77 

By  a  significant  figure  is  here  meant  any  of  the  ten  digits 
except  such  zeros  as  are  inserted  merely  to  enable  the  decimal 
point  to  be  located.  Thus  in  the  number  206.704  every  figure 
would  be  called  a  significant  figure  and  there  would  be  said  to 
be  six  significant  figures  or  six  places  of  significant"  figures.  In 
206.70400  there  would  be  nothing  to  show  whether  the  last 
two  zeros  were  significant  or  useless,  but  adopting  the  above 
definition,  as  will  always  be  done  in  these  notes,  they  would  be 
significant  figures,  of  which  therefore  there  would  be  eight.  In 
206  700.  there  would  be  nothing  either  in  usage  or  form  to  indi- 
cate whether  the  last  two  zeros  were  significant,  i.e.  whether 
there  were  four,  or  five,  or  six  significant  figures.  A  direct  state- 
ment by  means  of  the  precision  measure  or  otherwise  would  be 
necessary  in  such  a  case.  In  o.oo  206  7,  the  three  zeros  pre- 
ceding the  2  would  not  be  significant  and  there  would  be  four 
significant  figures.  In  o.oo  020  670  there  would  be  five  signifi- 
cant figures  under  the  usage  of  the  following  rules. 

It  is  clear  that  the  number  of  significant  figures  is  in  no  way 
determined  or  influenced  by  the  position  of  the  decimal  point. 
That  point  is  merely  an  arbitrary  mark  to  show  that  the  place 
before  it  is  the  place  of  units. 

When  either  phrase  "  number  of  significant  figures"  or 
"  number  of  places  of  significant  figures"  or  "  number  of  places 
of  figures"  or  "  place  of  figures"  or  "  place"  is  here  used,  it  will 
be  understood  to  mean  as  above  explained  and  to  have  no 
reference  whatever  to  the  number  of  decimal  places. 

This  definition  of  the  term  "  significant  figure"  is  not  in 
accordance  with  that  sometimes  given  which  limits  the  mean- 
ing to  the  nine  digits  other  than  zero.  But  zero  when  not  used 
merely  to  locate  the  decimal  is  just  as  significant  as  any  other 
digit.  For  any  digit  is  significant  only  in  the  sense  that  it 
shows  the  amount  of  the  stated  quantity  which  exists  in  the 
place  where  the  figure  stands,  and  a  zero  just  as  truly  denotes 
the  amount  that  exists  there  as  any  other  digit  would  do. 

Rules  for  Significant  Figures. — Let  m  denote  any  numer- 
rical  result  of  a  measurement  either  direct  or  indirect,  being 
either  a  single  observation  or  a  mean  result.  Let  $  denote 


78  INDIRECT  MEASUREMENTS, 

either  its  deviation  measure  or  its  precision  measure,  both 
based  upon  the  average  deviation,  viz.,  on  the  a.d.  if  m  is  a 
single  observation,  on  the  A.D.  if  m  is  a  mean. 

Rule  1. — In  casting  off  places  of  figures,  increase  by  I  the 
last  figure  retained,  when  the  following  figure  is  5  or  over. 

Example  V. — Thus  12.547  becomes  12.55  if  one  place  is 
rejected,  and  12.5  if  two  are  cast  off. 

If  the  figure  in  the  last  place  to  be  retained  happens  to  be 
zero  it  should  not  be  omitted  on  that  account,  but  the  zero 
should  be  written  in  just  as  any  other  digit  would  be,  to  show 
that  the  quantity  has  been  measured  or  the  result  carried  out 
to  that  place. 

Example  VI. — If  one  place  is  to  be  rejected  from  43.704  it 
would  be  written  43.70,  not  43.7.  See  also  similar  instances 
under  other  rules. 

Rule  2. — In  the  £,  retain  two  and  no  more  significant 
figures. 

Example  VII. —  6  =  5200.,     <?  =  0.085,     *  =  3-O. 

Rule  3. — In  the  quantity  m  retain  enough  significant  figures 
to  include  the  place  in  which  the  second  significant  figure  of 
the  d  occurs. 

Example  VIIL— Thus 


m  =  124.032  becomes  124.0 
6  —       1.4  1.4 


m  =  762  385.  becomes  762  390. 
6  =        680.  680. 


Quantities,  m,  which  are  obtained  by  a  single  observation 
often  fall,  unavoidably,  one  place  short  of  this  requirement. 
Example  IX. — 

m  =  1.875 
3  —  0.0038 

Ride  4. — When  several  quantities  are  to  be  added  or  sub- 
tracted. 


RULES  FOR   SIGNIFICANT  FIGURES.  79 

Find  the  quantity  which  has  the  largest  £,  (not  d/m,)  and 
retain  in  it  figures  as  under  Rule  3.  In  each  of  the  other 
quantities,  strike  off  all  significant  figures  beyond  that  corre- 
sponding to  the  last  place  retained  in  the  one  having  the 
largest  #. 

Example  X. — 

Data.  Computation. 

No.  p.m. 

23.85  0.26 

3  896.0  3.0 

-  4.23  37  0.00  47 

3915.7 

Rule  5. — When  several  numbers  are  to  be  multiplied  or 
divided  into  each  other: 

Find  by  inspection  the  one  for  which  d/m  is  greatest 
Compute  its  percentage  precision  100  d/m.  If  this  is 

I  per  cent  or  worse,  use  four  significant  figures,* 
"      "      "       "     ,    "    five  "  "      , 

«      «       "     ,    "    six  "  «      , 

in  all  data,  constant  factors,  products,  quotients,  and  in  the. 
result. 

When  the  A/M  of  the  result  is  computed,  any  places  in  M 
retained  under  the  above  rule  may  be  rejected  if  they  exceed 
the  requirements  of  Rule  3. 

Example  XL — To  be  multiplied  together:  29.34  with  8  = 
0.58,  42  231.6  with  d  •=.  1.4,  and  ^  =  3.14159265.  The  first 
number  is  clearly  the  least  reliable.  Its  percentage  deviation 
is  0.58  -T-  29.  —  2.0  per  cent.  Hence  we  should  use  four  sig- 
nificant figures  throughout. 




*  It  is  understood,  of  course,  that  this  means  if  100  —  3  i  per  cent  and  <  IO 

m  ^ 

oer  cent,  and  so  on  for  the  others. 


8o 


INDIRECT  MEASUREMENTS. 


Ordinary  Multiplication. 

29.34                  1239000 
42230               3.142 

Shorten 

29-34 
42  230. 

8802 
5868 
5868 
11736 

2478 
4956 
1239 
3717 

11736 
586 
58 
9 

1239028.2 

3892938. 
3  893  ooo. 

i  239000. 

I  239000. 

3-142 

3717 
1239 
496 

24 

3  893  ooo. 


Logarithms. 

29.34  1.4675 

42  230.  4.6256 

3.142  0.4972 


3  893  ooo. 


6.5903 


In  this  shortened  multiplication  the  partial  products  cannot 
safely  be  omitted,  until  one  place  beyond  that  to  be  retained 
is  reached.  The  method  is  sufficiently  obvious  upon  inspec- 
tion. It  saves  about  one  third  of  the  time  required  for  such 
work. 

Rule  6. — When  logarithms  are  used,  retain  as  many  places 
in  the  mantissae  as  there  are  significant  figures  retained  in  the 
data  under  Rule  5.  The  characteristic  is  not  to  be  considered, 
as  it  serves  merely  to  locate  the  decimal  point. 

Example  XII. — See  under  Rule  5. 

Demonstration  of  Rules. — Rules  i,  2  and  3.  The  reasons  for 
these  have  already  been  given  at  page  18.  Briefly  stated  they 
are  these :  Let  the  place  of  m  corresponding  to  that  of  the 
second  significant  figure  in  its  tf  be  called  the  rth  place.  As 
m  is  uncertain  by  an  average  amount  of  ±  #,  any  change 
which  is  ^-jV^  mav  be  neglected.  The  smallest  value  of 

will  be  i  in  the  rth  which  will  occur  when  8  =  10  in  the 


RULES  FOR   SIGNIFICANT  FIGURES.  8 1 

rth  place.  Then  we  may  reject  all  figures  in  m  which  do  not 
affect  it  beyond  I  in  the  rth  place.  From  the  rejection  of  all 
places  beyond  the  rth  the  error  in  m  will  not  exceed  0.5  in  the 
rth  place  if  Rule  I  be  followed.  Hence  all  places  beyond  the 
rth  should  be  rejected  as  called  for  by  Rule  3.  The  d  is  re- 
tained to  two  significant  figures  chiefly  because  in  computing  a 
AM  from  tf, ,  #2 ,  etc.,  or  vice  versa,  it  is  essential  to  keep  the 
first  place  surely  correct. 

Rule  4.  Let  m'  denote  that  one  of  the  given  quantities  to  be 
added  or  subtracted  which  has  the  largest  precision  measure 
or  deviation  measure,  6f,  based  on  the  average  deviation ;  e.g., 
the  number  3  896.0  in  Example  X.  Let  the  place  of  figures 
(not  of  decimals)  in  m  corresponding  to  the  second  significant 
figure  in  d'  be  called  the  rth.  Then  it  is  clear,  as  in  the  ex- 
ample, that  most  if  not  all  of  the  figures  in  the  other  quantities 
which  are  in  places  beyond  the  rth  of  m'  are  useless.  The 
question  is  then  just  what  figures  may  be  stricken  out. 

Let  n  be  the  number  of  data,  some  of  which  may  be  con- 
stants, then  the  greatest  number  of  rejections  to  be  made  will 
be  n.  Suppose  then  that  we  follow  Rule  4,  making  n  rejec- 
tions by  striking  out  all  figures  beyond  the  rth  place  of  m' ; 
what  will  be  the  accumulated  rejection  error  in  the  result  ? 

The  law  of  distribution  of  the  rejection  errors  is  the  special 
law  of  deviations  given  at  page  2 1 .  For  we  never  reject  beyond 
the  limits  of  -|-  5  and  —  5  in  the  r  -f-  1st  place,  and  any  value 
between  these  limits  is  equally  likely  to  occur  at  any  rejection. 
The  average  error  for  a  single  rejection  is  then  one  half  of  this 
limit,  viz.,  2.5  in  the  r  -\-  1st  or  0.25  in  the  ?-th  place.  Con- 
trary to  the  usual  case  the  sign  of  the  rejection  error  is  known 
for  every  rejection,  and  its  magnitude  also  to  at  least  one 
place  of  figures.  So  that  if  we  chose  we  could  compute 
exactly  the  actual  accumulated  rejection  error  in  any  given 
problem.  But  as  we  do  not  care  to  do  this,  but  wish  to 
deduce  a  rule  which  shall  be  safe  to  use  in  all  cases,  we  must 
assume  the  accumulation  to  be  according  to  the  formula  [24], 
viz., 


82  INDIRECT  MEASUREMENTS. 

where  E  is  accumulated  error  of  result,  and  el ,  et  the  errors  at 
each  rejection.  Then  if  e  is  the  average  rejection  error,  viz., 
0.25  in  rth  place,  we  may  write 

E1  =  tie*  approx. 

This  assumes  that  e*  =  (e*  -j-  . . .  +  en*)/n,  which  is  a  suffi- 
ciently close  approximation  under  these  conditions.  Hence 

E  —  e  Vn  =  0.25  Vn  in  rth  place. 

Strictly,  mean  e>=  E*  =  *  _  Ja  f(*)dx,  where  f(x)  •=  x*  (since 
the  frequency  is  constant)  and  x  =  e,  a  =  o,  b  =  0.5 ; 

E>-—  °'6--L  E*  -  nf  -  —          E-  \/~^- 

'     3     0    ""  12*  "12*  72~' 

Thus  if  n  =  12,  E  =•  I  unit  in  rth  place,  which  is  negligible  for 
the  worst  value  of  J/as  shown  in  demonstration  for  Rules  i, 
2,  and  3.  Now  n  would  rarely  be  as  great  as  12,  so  that  this 
rule  is  sufficiently  exact  for  the  worst  case,  and  therefore  more 
than  close  enough  for  any  case  of  ordinary  practice. 

It  should  be  noted,  however,  that  E  as  thus  computed  is 
not  a  definite  quantity.  It  is  merely  an  average  value  of 
E.  But  as  it  is  calculated  on  the  same  basis  as  the  precision 
measure  6  or  A,  and  is  therefore  a  quantity  of  the  same  nature, 
it  is  proper  to  use  -fad  or  -faA  in  fixing  its  limit. 

Rule  5.  This  rule  may  be  justified  as  follows  for  the  case 

8' 
where  — 7  >    I  per  cent.     A  similar  proof,  of  course,  applies 

to  all  cases. 

^  o 

The  -=&  of  the  result  will  be  c:  — /  of  the  least  fractionally 
Jyl  ^  m 

precise  quantity,  by  formula  [52],  so  that  -jjr  =  i  per  cent. 
Therefore j  and  —  j^  will  be  negligible.  The  smallest 


RULES  FOR   SIGNIFICANT  FIGURES.  83 

value  of  A  corresponding  to  this  limit  will  be  ±  I  in  the  fourth 
place,  viz.,  when  M  —  1000  ;  and  we  must  not  allow  the  ac- 
cumulated computation  error  to  exceed  this  amount.  This 
will  be  accomplished  under  the  Rule  5. 

The  worst  case  under  this  rule  is  when  the  first  four 
figures  in  each  factor  of  the  data,  constant,  intermediate  prod- 
uct or  quotient,  and  the  final  result,  are  1000.,  and  if  it  can  be 
shown  that  for  this  the  accumulated  rejection  error  does  not 
exceed  the  limit,  the  rule  will  be  justified. 

Let  m^  ,  m^,  .  .  .  ,  mn  denote  the  values  of  the  various  data, 
constants,  and  intermediate  products  or  quotients,  and  suppose 
a  rejection  to  be  made  at  each  of  them,  viz.,  n  rejections  in  all, 
leaving  errors  el  ,  e^  ,  .  .  .  ,  en  .  Then 


But  when  expressed  in  units  in  the  4th  place,  we  have  under 
the  above  conditions 

0 

m^  =  m^  =  .  .  .  =  mn  =  1000.  =  M, 


also  expressed  in  units  in  the  4th  place.  Therefore  as  under 
Rule  4,  for  n  —  12 

E  =  ±  i  in  4th  place. 

Thus  the  rule  is  justified  for  the  worst  case,  and  is  therefore 
sufficient  for  all. 

Here  also  it  is  to  be  remembered  that  E  as  thus  computed 
is  not  a  definite  but  only  an  average  value,  and  is  thus  a 
quantity  of  the  same  character  as  #  or  ^/,  and  the  criterion  of 

---  as  a  limit  of  negligibility  is  properly  applied.     It  is  not 

asserted  that  the  accumulated  rejection  error  will  never  exceed 
E  =  i  in  4th  place.  On  the  contrary  in  12  rejections  an 
error  E  of  6  in  the  4th  place  would  occur  if  all  the  errors  e 


84  INDIRECT  MEASUREMENTS. 

happened  to  be  of  the  maximum  value  and  in  the  same  direc- 
tion,— an  event  which  would  be  exceedingly  rare. 

Rules  which    provided   that    the    maximum    error   should 

i    z/ 
never  exceed  -    -jr=:  would  be  needlessly  wasteful  of  labor.    The 

limit  used  above  is  in  accordance  with  that  used  elsewhere  in 
this  book,  and  is  abundantly  close  for  all  work  except  possibly 
such  as  conforms  with  extraordinary  closeness  to  the  premises 
upon  which  the  method  of  least  squares  is  built  up — far  more 
closely  than  any  ordinary  physical  or  technical  work. 

Rule  6.  Inspection  of  a  table  of  logarithms  will  show  that 
in  the  worst  case,  i.e.,  where  m  =  9999,  a  change  of  I  unit  in 
the  4th  place  of  the  logarithm  will  correspond  to  a  change  of 
2  units  in  the  4th  place  of  the  number,  which  shows  that  the 
accumulated  error  will  be  about  twice  as  great  in  the  worst 
case  by  logarithms  under  rule  5  as  by  numbers  under  rule  4. 
But  on  the  average  the  error  is  very  nearly  the  same  by  loga- 
rithms as  by  numbers  for  the  same  value  of  n,  with  the  advan- 
tage in  favor  of  logarithms  as  reducing  n  by  requiring  fewer 
intermediate  'operations. 

A  more  complete  discussion  of  this  case  is  given  in  the 
author's  "  Computation  Rules  and  Logarithms."  The  other 
rules  also  are  there  discussed  in  a  different  manner. 

Forms  of  Problems  on  Accuracy  of  Result. — These  are 
the  same  as  those  given  for  direct  measurements  at  page  33  et 
seq.  The  procedures  there  outlined  apply  to  each  of  the  direct 
measurements  from  which  the  indirect  result  is  made  up.  It 
is  therefore  only  necessary  to  add  the  following. 

In  an  indirect  measurement  any  consideration  of  the  accu- 
racy of  a  result  involves  that  of  each  component  direct  measure- 
ment and  also  involves  the  discussion  of  the  possible  "  error  of 
method  "  (page  46)  of  the  process. 

The  problems  present  themselves  in  these  forms. 

First.  To  obtain  by  a  proposed  method  the  most  accurate 
result  practicable.  For  this  purpose  the  method  as  a  whole, 
and  the  method,  apparatus  and  conditions  of  work  in  measur- 
ing each  component,  must  be  studied  as  described  at  page  33. 


PLANNING  OF  INDIRECT  MEASUREMENT.  85 

Second.  To  obtain  a  measurement  of  the  desired  quantity, 
and  have  the  result  accurate  within  a  specified  limit.  For  this 
the  statements  made  at  page  34  apply  verbatim,  both  to  the 
whole  method  and  to  the  separate  components.  It  is  necessary 
only  to  add  that  a  preliminary  solution  by  equal  effects  should 
be  made  to  determine  approximately  the  best  value  of  the 
precision  measure  of  each  component.  Also  at  the  close  of  the 
actual  observations,  it  is  essential  to  make  a  final  calculation 
of  the  precision  measure  of  the  result  and  to  combine  with  this 
an  estimate,  so  far  as  one  is  possible,  of  the  "  error  of  method  " 
and  of  the  effect  of  any  constant  which  may  be  suspected  to 
exist. 

Third.  Given  a  completed  result  obtained  by  a  stated  method, 
to  estimate  its  accuracy.  The  remarks  of  the  corresponding 
section  at  page  35  apply  equally  to  indirect  measurements. 

Data  required  to  Substantiate  Result. — The  remarks  of 
page  36  apply  equally  to  indirect  measurements. 

Planning  of  Indirect  Measurement. — To  the  statements 
made  under  (a)  to  (d)  on  page  37  it  is  only  necessary  to  add 
that  a  preliminary  precision  discussion  based  on  approximate 
data  should  invariably  be  made  before  a  choice  of  methods  is 
finally  reached,  or  at  least  before  any  experimental  work  beyond 
preliminary  trials  is  entered  upon.  The  weak  or  strong  points 
of  the  proposed  methods  are  often  thus  developed,  and  impor- 
tant modifications  suggested  or  errors  avoided.  It  is  also  of 
great  importance  that  the  plan  for  the  "  reduction,"  i.e.,  the 
algebraic  and  numerical  calculations,  be  thoroughly  developed 
in  advance  of  the  measurements.  A  slight  modification  of  a 
proposed  method  may  sometimes  transform  the  reduction  from 
a  laborious  to  a  much  easier  one. 


86  INDIRECT  MEASUREMENTS. 


EXAMPLES. 

Example  XIII  (see  page  50).  —  The  value  of  g  is  to  be 
measured  by  a  simple  pendulum  whose  time,  /,  of  a  single 
vibration  is  to  be  about  2  seconds  ;  (a)  what  change  (in  units) 
in  g  would  correspond  to  or  be  produced  by  a  change  of 
o.i  cm.  in  /? 


Here  £•  corresponds  to  M,  I  to  m\  ,  t  to  ;#„  ,  and  TT*  is  a  constant. 
We  have  to  find  the  value  of  ^,  corresponding  to  dl  =  o.i  cm. 
By  [19] 


cm. 

A    A  =  2.4X0.  1  =  0.24^. 

(#)  What  change  (in  units)  in  g  would   correspond   to  a 
change  of  0.02  sec.  in  /  ? 

df(}-       27t*--       2y<i' 
~dt'       ~27t  f~    -  2  x  3.1 

the  length  of  a  2-sec.  pendulum  being  about  400.  cm. 

cm. 
.-.  J,  =  —  960  X  0.02  =  —  19.  -  5. 

The  negative  sign  indicates  that  a  +  change  in  /  produces  a 
-—  change  in  g. 

Example  XIV  (see  page  50).  —  In  a  measurement  of  g  by  a 
2-seconds  pendulum,  as  in  Example  XIII,  (a)  what  change  in  / 

would  produce  a  change  of  i.o  -  -,  in^-? 

sec. 


EXAMPLES.  87 

By  [21] 


.•.  61=  1.0/2.4  =  0.42  cm. 

(b)  What  change  in  /  would  produce  a  change  of  i.o  per 
cent 


i.o   per  cent  of  g  is  o.oi^,  and  as  g  =  980.  -  -a   nearly, 

sec* 

cm. 
=aoi  X  980.  =  9-8j^ri- 

By  [22] 


.•.  <J/  =  9.87  —  960.  =  o.oio  sec.  which  is  —  --  =  0.005  or  i 

per  cent  of  /. 

Example  XV  (see  page  53).  —  Suppose  that  in  Example 
XIII  the  changes  dt  =  o.i  cm.  and  6t  =  0.02  sec.  were  of  the 
nature  of  deviations  and  occurred  simultaneously,  what  would 
be  the  resultant  effect  on^-? 

By  [24] 


and  by  Example  XIII  4  =         .  */  =  0.24  —  3  ,  and 

sec* 


^/  sec. 

8  =  (o.24)2  +  (-  i9.)2  =  0.058  +  360.  =  360. 


88  INDIRECT  MEASUREMENTS. 

Example  XVI  (see  page  54). — In  the  measurement  of  g 
by  a  2-seconds  pendulum,  as  in  the  foregoing  examples,  what 
changes  81  in  /  and  6t  in  /  would  have  a  combined  effect  on  g 

of  Ag  =  3.0 '-? ,  supposing  them  to  be  of  the  nature  of  devia- 

sec. 

tions  ? 

Solving  for  equal  effects  by  [29]  etc.,  we  have 


and  *=-. 

/Y    dl  /    <*t 


g 

~j  —  2.4,     -~  =  —  960.,  as  before,     and     A  =  3.0 


.x     _  =  0.88  cm. 
1.4      2.4 


Example  XVII  (see  page  57). — The  rise  of  temperature 
A  =  t^  —  /,  of  the  water  of  a  continuous  calorimeter  is  to  be 
measured  by  the  reading  of  two  thermometers. 

(a)  What  will  be  the  precision  of  A  as  affected  by  /,  alone 
if  the  p.m.  dtl  is  O°.O2  ? 

By  [40], 

A  =  ±  *i- 

/.   A,  =  O.°02. 

(b)  What  will  be  the  p.m.  of  ^4  for  a  /.*#.  of  d/,  —  o°.O2  and 
<j/f  =  o°.03  ? 

By  [42], 

^a  =  0.02'  +  0.03*  =  o.oo  04  +  o.cx>  09  =  o.oo  13. 
J  =  o°.o36. 


EXAMPLES.  89 

(c)  What  will  be  the  p.m.  necessary  in  each  component  for 
a  p.m.  of  o°.oi  in  A? 

By  [43], 

o.oi 

dt.  =  St.  =  —=  =  0°.007I. 

V2 

Example  XVIII  (see  page  58). — An  incandescent  lamp 
burns  for  a  time  t  under  a  voltage  v  and  with  a  current  c.  The 
quantities  c,  z/,  and  /  are  measured  in  order  to  determine  the 
amount  of  heat  H  produced  in  this  time. 

H=kcvt, 
where  k  =  a  constant. 

(a)  What  p.m.  in  H  would  correspond  to  a  precision  of  o.  i 
per  cent  in  c? 

6c 

—  =  o.i  per  cent  =  o.ooi. 
c 

A,       dc 
By  [50]  —=—=  o.ooi. 

AH 
The  fractional  precision  in  H  would  then  be  -jj-   =  O.OOI    or 

o.i  per  cent,  whence  AH  =  o.ooi// could  be  found  if  desired, 
if  H  were  known. 

(b)  What  percentage  precision  in  v  alone  would  correspond 
to  0.5  per  cent  in  H? 


—  =  ]jf  =0.5  per  cent. 

(c)  What  would  be  the  fractional  precision  in  H  resulting 

dc  dv 

from    a     fractional     precision    of    —  =  O.OOI,    - —  =0.003, 

dt 

—  =  0.002  in  the  components  ? 

/  J  V 

\M  /   =  aooia  +  o-oos8  +  0.002'  =  o.oo  ooi  4 ; 

^  =  0.00  37. 


90  INDIRECT  MEASUREMENTS. 

(d)  What  percentage  precision  in  each  component  would 
correspond  under  equal  effects  to  0.2  per  cent  in  HI 

dc       Sv       dt         I 

—  =  —  =  —  =  -—  X  O.002  ==  O.OOI2, 
c         v         t        4/3 

or  o.i  2  per  cent  in  each. 

Example  XIX  (see  page  59).  —  The  volume  of  a  sphere  is 
to  be  computed  from  its  measured  diameter  D.      V= 


(a)  If  the  precision  of  D  is  I  per  cent,  what  is  the  precision 
of  th 


e  result  ? 


A          d 

—  =  v-^  =  3  X  o.oi  =  0.03,  or  3  per  cent. 


(b)  What  precision  would  be  requisite  in  D  for  I  per  cent 
in  F? 

6      i   A 

-  =  -  =pr  =  i  X  o.oi  =  0.0033  or  J  per  cent. 

Example  XX  (see  page  60).  —  For  the  sake  of  comparison 
with  former  examples  take  the  case  of  the  measurement  of  g 
with  a  2-sec.  pendulum. 

(a)  What  would  be  the  precision  of  the  result  if  the  frac- 
tional precision  of  /  were  o.i  per  cent,  and  of  /  the  same? 


We  should    separate   into  the   factors  n'  x  /  X  t".     Hence 
by  (a) 


=  0.00  ooo  i  +  o.oo  ooo  4  =  o.oo  ooo  5, 

A 

—  =  o.oo  22,     or    0.22  per  cent. 


EXAMPLES.  91 

(b)  What  would  be  the  values  of  to  and  to  necessary  under 
equal  effects  to  give  g  with  a  precision  of  o.i  per  cent? 
By  [67] 

61         6t        i 

->-  =  2—  —  —7-  x  o.ooi.  =  o.oo  071. 

/  t          4/2 

d/ 

.*.  -y  =  0.00071,     or    0.071  per  cent, 


—-  =  -  x  o.oo  071  =  0.00  036,  or  0.036  per  cent 

/  2 

To  find  SI  and  to,  we  must  know  /  and  t.     t  is  stated  to  be  2 

sec.,  and  as  g  must  be  about  980  --  a,  /must  be  about  4.001. 

sec 

/.  M  =  o.oo  071  X  400.  =  0.28  cm. 
dt  =  o.oo  036  X  2.0  =  0.00071  sec. 

Example  XXI  (see  page  62).  —  The  measurement  of  a  cur- 
rent by  a  cosine  galvanometer  affords  a  good  example  of  a 
function  which  can  be  separated  into  the  product  of  several 
functions,  each  of  one  component  only.  For  a  primary  cosine 
galvanometer  the  formula  may  be  written 


tan  0 
C=  ----  - 

27TH       COS  GO 


where  H=honz.  comp.  of  earth's  field,  r  =  mean  radius  of 
coil,  n  =  whole  number  of  turns  in  coil,  0  =  angle  of  deflec- 
tion of  needle  under  the  current  C,  GO  =  angle  of  tip  of  coils 
from  vertical  when  0  is  read.  The  function  would  be  separated 
as  follows : 

_  10  I  I 

~  27Z  n        '  cos  GO' 


92  INDIRECT  MEASUREMENTS. 

Of  the  components,  H  and  r  enter  as  simple  factors,  n  as  a 
factor  to  the  —  i  power,  0  in  the  function  tan  0,  and  GO  in  the 

function  -    or  (cos  coY1  ,  both  of  these  functions  being 

COS  63 

factors.  For  this  value  of/(),  the  fractional  precision  would 
be,  by  [69], 

_f*H\*      I6r\*  ,   /*r'y  ,   /*tan_0\'      f^cos^-y 
:=  ny/  "h  V7V  -r  V1F/  '  -  V  tan  0  J  +  \(cos  a.)'"  /  ' 

Suppose  a  measurement  in  which  the  precision  of  the  compo- 

dH  $r  dn 

nents  was  jy-  =  o.ooi,  —  =  0.000$,  —  =  negl,  d0  =  o  .025, 

4 
6cj  =  o°.O25,  what  would  be  the  value  <tf  Tf  ? 

From  [61]  we  know  that  —  r  =  —  —  .     We  require  further 

n~l  n 

the  values  of  -  -7-  and  -  .    To  find  these  we  must  have 
tan  0  cos  GO 

the  values  of  0  and  GO.  The  data  would  furnish  this,  or  in  a 
preliminary  discussion  typical  or  limiting  values  would  be  used. 
Suppose  0  =  45°  and  GO  =  60°.  Then  by  [33] 


tan  0  =  •  £0  =  sec3  0 


d  tan  0  _  sec8  0         _     2(?0 
tan  0    '  ~  tan  0  ~  sin  20  ' 


To  compute  the  numerical  value  of  this  we  must  have  #0  ex- 
pressed in  terms  of  n. 

7t 

o°.o25  =  0.025  X  -TT-  =  0.025  X  0.017  =  o.oo  043. 

Hence 

6  tan  0      2  X  o.oo  043 

—  = : Ep*   =  0.00  086. 

tan  0  sin  90 


EXAMPLES.  93 

Also 

#(COS  ft?)'1  _  8  COS  Ca? 

(cos  a?)'1  "  cos  GO  ' 

d  cos  GO  =  -7-  (cos  ca)  •  #o>  =  —  sin  <»  *  #<» 
^ 


cos  G?      sn  GO 

=  -  •  dco  =  tan  GO  •  OGO. 

COS  GO  COS  GO 


Substituting  gives 

£(cos  ft?)'1 
(cos  <a)rr  =  I-7  X 

Then 

(^\2 
-^  )  =  o.oo  i8  +  0.0005'  +  o3  +  o.oo  o86a  +  o.oo  073' 

=  o.oo  ooo  25. 
g)  =  0.0016. 

Equal  Effects.  —  What  fractional  precision  in  each  compo- 
nent would  be  necessary,  6n  being  negligible,  in  order  that  A/M 
should  be  0.2  per  cent? 

For  equal  effects  we  must  have 

dH      dr  _  dn~l  _  6  tan  0  _  <?(cos  a?)"1         I     A 
~H=~r~'~n-1~      tan0         (cos  a?)'1  ""  y^M' 


As  before — -  = —,     and     ~ r^r  =  tan  GO -Sao. 

tan  0        sin  20  (cos  ft?) 


94  INDIRECT  MEASUREMENTS. 

The  number  of  components  is  n  =  4.      .*.  —-=.-jr=  =  %  X  0.002 


=  O.OOIO. 


SH 
/.  -77^  =  o.ooio ; 


—  =  o.ooio' 


#  tan  0 

=  O.OOIO  =  — 


tan  0  sin  20* 

/.  d0  =  0.00050  sin  90°  =  0.00050; 


T  =  o.ooio  =  tan  GO.  do?. 
~ 

&  ==  o.ooio/tan  60°  =  0.00060. 

o         0.0006O 


0.017 


_  0°.035. 


Example  XXII  (see  page  65).  —  The  following  is  taken  as 
a  simple  illustration  of  the  separation  into  groups.  More 
complex  examples  will  occur  in  connection  with  the  tangent 
galvanometer,  etc. 

A  continuous  water  calorimeter  is  to  be  tested  by  trans- 
forming into  heat  within  it  a  measured  amount  of  electrical 
energy  and  measuring  this  heat  by  the  calorimeter.  For 
instance  some  incandescent  lamps  or  a  coil  of  wire  carrying  a 
current  are  placed  within  the  calorimeter,  the  mean  current  c 
through  the  coil  and  the  mean  voltage  v  at  its  terminals  are 
measured  ;  also  the  mass  m  of  water  passing  through  the 


EXAMPLES.  95 

calorimeter  during  the  measured  time  r,  and  the  temperatures 
/,  and  /,  of  the  entering  and  outflowing  water.     Then 


[86] 


where  k  is  a  constant,  viz.,  the  heat-equivalent  of  I  watt, 
calculable  from  the  mechanical  equivalent  of  heat  and  of  the 
watt.  The  test  consists  in  ascertaining  how  closely  the  experi- 
mental value  of  £,  viz., 

k  =  m&  ~  *») 
cvt 

agrees  with  the  computed  value. 

What  precision  is  requisite  in  each  component  for  a  test  to 
o.i  per  cent? 

The  problem  may  be  solved  by  the  general  formula,  but 
could  not  be  solved  exactly  by  the  simple  formula  for  factors, 
since  one  factor  (/a  —  /,)  contains  two  components.  Applying 
the  formula  [77]  for  separation  into  groups,  we  have  for  equal 
effects 


™         4/2    t*  -  v-         r~ 

Noting  that  —-^-  =  —  —  ,  etc.,  and  neglecting  signs  as  of  no 
c  c 

consequence,  we  have 

dm       dc      dv       dr         i 

—  =  —  =  —  =  —  =  —  X  O.OOI  =  O.OO  040, 

m        c         v         r        2.5 

from  which  the  numerical  values  of  6m,  dc,  dvt  and  6r  could 
be  computed  if  m,  c,  v,  and  r  were  given. 

-^  —  -J-  =  4/2  X  0.00040  =  0.00056. 

7    —  * 


96  INDIRECT  MEASUREMENTS. 

To  find  dtl  and   tf/3  we   must  have   the  value  of 
Suppose  this  to  be  given  as  10°,  then 

<?(/„  —  /J  =  o°.oos6. 
Now  by  [43],  for  equal  effects 


-~  *(/,  -  /,)  =  o°.oo40. 


Example  XXIII  (see  page  68). — On  a  cradle  dynamometer 
or  on  a  friction  brake  the  horse-power  is  given  by 

27tRNW 


where  ^  is  the  radius  at  which  the  load  W  is  applied  and  N  is 
the  number  of  turns  per  unit  of  time.  N  really  involves  two 
measured  components,  viz.,  a  time  and  a  count  ;  but  the  count 
is  usually  made  by  mechanical  means,  in  such  a  way  that  the 
error  of  observation  resides  wholly  in  the  time.  We  might 

therefore  well  substitute  for  N  its  equivalent  expression    ~, 

where  T  is  the  time  of  a  single  rotation,  and  thus  find  #7".     But 

we  may  just  as  well  proceed  to  find  dN  and  from  that  find  6T. 

(a)  Suppose  it  required  to  find  whether  with  no  other  rejec- 

dW  A 

tion  --  =  ^th  per  cent  would  be  negligible  for  -     =  0.001. 


The  negligible  limit  by  the  criterion  would  be,  as  before, 
3W  _i    A 


and  ^th  per  cent  =  0.0005  would  not  be  negligible. 


EXAMPLES.  97 

dn  dW 

(b)  Suppose  it  required  to  find  what  values  of  —  and  -f^ 

7t  W 

A 

would  be   simultaneously  negligible  for  -JT=  =  0.001.  By  the 

criterion  this  would  be  when  either  was 


I     i      d        ii 

-  —^-rf  = 7=  X  o.ooi  =  0.00024. 

3    Vp  M       34/2 


(c)  Suppose  the  problem  to  be,  how  far  must  the  constant 
-  be  carried  out  in  order  that  the  rejection  error  shall  be 


33000 

negligible  with  respect   to     „  p     =  0.00 1.   .  The   rules   for 

significant  figures  would  require  it  to  be  carried  to  5  places, 
and  as  n  is  the  only  term  in  it  in  which  a  rejection  would  be 
made,  n  must  be  carried  to  5  places  of  significant  figures,  viz., 
to  3.1416.  Let  us  apply  the  rule  of  this  section  and  see 
whether  a  similar  result  will  be  reached. 

As  only  one  rejection  is  to  be  made,  and  as  n  enters  as  a 

dn 
direct  factor,  we  shall  have  —  negligible  when 

dn  _  i   A 


or 

dn  =  0.00037T  =  o.oo  093. 

Carrying  n  only  to  3.142,  the  rejection  only  makes  an  error 
of  tf TT  =  0.0004,  which  is  within  the  limit  assigned  by  the 
criterion.  Hence  by  following  the  criterion  we  should  use 
TT  =  3.142,  by  the  rules  for  significant  figures  n  =  3.1416,  so 
that  if  the  criterion  is  reliable  the  rules  are  more  than  suffi- 
ciently precise  in  this  case. 


98  INDIRECT  MEASUREMENTS. 

f 

(d)  Suppose  the  question  to  be,  would  the  use  for  n  of 

dW 
3.142  be  admissible  with  a  precision  of  -TFF  =  O.OO  030  and  a 

precision  of  O.I  per  cent  required  in  the  result? 
By  the  criterion  this  would  be  the  case  if 


*  -  i  A 

=0.00033; 


dn      o.oo  041 

-  =  —  -=0.00013; 

•'•  Vetc.  =  0.00033. 

This  would  be  barely  negligible. 

Example  XXIV  (see  page  76).  —  The  expression  for  the 
specific  resistance  per  metre-gramme  at  o°  c.  of  a  wire  may  be 
written  in  the  form 

s_m       ,     i+firT          I 
-  /'  j  4-  ftrr     i  +  fit? 

where  m  —  mass,  /  =  length,  rr  =  resistance  of  the  sample 
measured  on  a  Wheatstone  bridge,  ft'  =  its  temperature  coef- 
ficient, /  =  its  temperature  at  the  time  of  measurement,  ft  = 
temp,  coeff.  of  bridge,  T  =  observed  temperature  of  bridge, 
r  =  temperature  at  which  it  is  correct. 

For  the  precision  discussions  this  expression  may  be  simpli- 
fied by  using  the  approximations 


=  l-0rT  approx.     and 
These  would  leave  the  expression  in  the  form 

S=£-.r'.(i  +ftTT)(i  -  ftrr)(i  -  ft/t)  approx., 


EXAMPLES.  99 

which  may  be  still  further  simplified  by  using  another  approxi- 
mation and  writing 


"  approx. 


These  approximations  are  well  within  the  limit,  for  the  terms 
/3T,  0r,  and  ft't  are  all  less  than  o.i,  so  that  the  error  of  the 
approximation,  which  is  smaller  still,  is  negligible  in  proportion 
to  I  in  the  parentheses. 

This  final  expression  affords  a  good  illustration  of  the 
method  of  separation  into  groups.  Counting  the  temperature 
coefficients  and  r,  as  we  should  do  in  a  preliminary  discussion 
at  least,  where  there  was  any  question  as  to  the  possibility  of 
their  being  known  accurately  enough,  we  have  in  the  [  ]  six 
components,  and  in  all  nine.  Hence  for  equal  effects  by  [78] 
we  should  have 


dm 


where  we  write  #[]/!  instead  of  #[]/[],  because  []  =  i  sensi- 
bly. The  precision  of  the  components  in  the  [  ]  can  thus  be 
much  more  conveniently  studied  than  by  using  the  general 
method  which  would  otherwise  be  necessary. 

The  omission  of  terms  in  differentiating  is  illustrated  in  the 
examples  on  the  tangent  galvanometer,  cradle  dynamometer, 
and  on  the  Stray  Power  Test  of  a  dynamo. 


BEST  MAGNITUDES  OF  COMPONENTS. 


Nature  of  Problems. — Another  class  of  problems,  quite 
distinct  from  those  which  have  been  discussed,  is  capable  of 
solution,  more  or  less  complete  according  to  circumstances,  by 
the  methods  which  have  been  developed.  These  methods  have 
been  applied  to  the  calculation  of  the  precision  of  the  result  of 
an  indirect  measurement  from  the  precision  of  its  components, 
and  to  the  determination  of  the  best  precision  of  the  various 
components  for  a  specified  precision  of  the  result.  Beyond 
these  it  often  happens  in  designing  apparatus  or  in  planning  the 
work  of  an  indirect  measurement  that  such  problems  as  the 
following  are  met.  Given  or  having  decided  upon  the  ap- 
paratus by  which  the  work  must  be  done,  and  thus  the  pre- 
cision with  which  the  components  can  be  measured,  it  is  found 
that  there  is  some  freedom  in  the  proportioning  of  the  parts  of 
instruments,  or  in  the  assignment  of  magnitude  to  some  of  the 
components ;  and  the  problem  arises  to  determine  what  will 
be  the  best  magnitudes  under  the  conditions  of  the  work. 
To  study  some  specific  problems,  suppose  that  the  resistance 
of  a  battery  is  to  be  measured  by  the  ordinary  two-deflection 
method,  using  a  tangent  galvanometer  which  reads  to  o°.i  by 
a  pointer  on  a  graduated  circle  ;  what  are  the  best  deflections 
to  use,  i.e.,  what  two  deflections  will  give  the  desired  result 
with  greatest  precision,  other  things  being  equal? 

Again,  suppose  there  is  to  be  designed  a  circular  cylinder 
of  brass,  whose  moment  of  inertia  around  a  transverse  central 
diameter  is  to  be  calculable  from  its  mass  and  measured  dimen- 

100 


NATURE  OF  PROBLEMS.  IOI 

sions,  both  diameter  and  length  being  measured  by  the  same 
instrument  and  with  the  same  p.m.  ;  what  will  be  the  best 
length  and  diameter  for  the  cylinder? 

Again,  what  is  the  best  angle  at  which  to  use  an  ordinary 
tangent  galvanometer  as  far  as  errors  of  reading  are  con- 
cerned ? 

The  following  statements  are  expressed  in  general  terms  for 
indirect  measurements.  Of  course  problems  concerning  the 
design  of  apparatus,  such  as  that  of  the  cylinder  for  moment 
of  inertia,  fall  directly  under  these  statements.  Thus  in  this 
example  the  problem  is  to  so  construct  the  bar  that  its  dimen- 
sions shall  be  the  best  components  in  the  indirect  measure- 
ment of  its  moment  of  inertia. 

The  possibility  of  such  control  of  apparatus,  method,  or 
magnitude  of  components  by  no  means  always  exists.  It  is 
frequently  precluded  by  the  nature  of  the  process,  or  by  the 
number  of  conditions  or  restrictions  placed  upon  the  work. 
For  instance,  if  the  efficiency  of  a  dynamo  running  under  stated 
conditions  is  to  be  measured  by  a  stated  electrical  method,  the 
magnitude  of  all  the  quantities  are  predetermined  by  the  con- 
struction and  capacity  of  the  machine,  and  by  the  stated  condi- 
tions of  the  test.  On  the  other  hand,  if  a  certain  quantity  of 
heat  is  to  be  produced  by  a  current  through  a  wire  and  this 
amount  is  to  be  computed  from  measurements  of  the  current, 
c,  resistance,  r  (or  potential),  and  time,  /,  there  may  be  best 
relative  magnitudes  of  c,  r,  and  t  for  a  given  set  of  measuring 
instruments.  It  is  however  less  common  to  meet  problems  on 
best  magnitudes  than  on  the  previous  parts  of  the  subject. 

The  problem  of  best  magnitudes  is  to  some  extent  a 
Reversal  of  that  of  best  precision  of  components,  but  not 
strictly  so.  The  data  given  are  the  precision  of  the  compo- 
nents, and  the  problem  is  to  determine  the  best  magnitudes ; 
and  to  this  extent  it  is  the  converse  of  the  other  proposition. 
But  we  have  no  longer  any  considerations  as  to  the  amount  of 
labor  involved  or  as  to  its  distribution,  for  that  is  determined 
by  the  precision  conditions,  which  are  now  fixed.  The  solu- 
tion consists,  then,  merely  in  finding  the  relative  and  numerical 


K 
102  BEST  MAGNITUDES  OF  COMPONENTS. 

values  of  some  or  all  of  the  components  which  will  make  A  a 
minimum  for  the  given  function  and  precision  conditions.  If 
the  components  are  independent  quantities,  we  may  solve  for 
the  best  magnitudes  of  all  of  them,  if  desired.  If,  however, 
some  or  all  of  them  are  conditioned,  e.g.,  if  one  is  a  function  of 
one  or  more  of  the  others,  the  solution  cannot  be  made  for  all 
of  those  thus  conditioned ;  one,  at  least,  of  the  conditioned 
quantities  must  be  free,  i.e.,  must  be  determinable  with  such 
a  precision  that  its  numerical  magnitude  may  be  anything 
which  may  be  required  by  the  best  values  of  the  others  by 
which  it  is  conditioned.  When  there  are  three  or  more  com- 
ponents, whether  independent  or  conditioned,  it  usually  occurs 
that  the  precision  conditions  for  one  or  more  are  such  that 
these  quantities  can  be  omitted  from  the  consideration,  that  is, 
that  their  magnitudes  may  be  anything  whatever  which  may  be 
required  by  the  others,  thus  simplifying  the  problem. 

For  a  single  component  the  problem  takes  this  form. 
Having  given  the  form  of  the  function /"(/«)  and  the  value  of 
6  or  d/m,  it  is  desired  to  know  the  best  value  of  m  for  a  given 
value  of  M. 

That  this  may  be  solvable,  there  must  occur  in  f(m)  some 
constant  or  other  quantity  which  can  be  changed  so  that  for 
any  value  of  m  the  value  of  M,  that  is,  of  f(m),  may  be  made 
of  the  specified  amount.  For  instance,  in  a  tangent  galvanom- 
eter C  =  K-  tan  0  represents  the  current  C  producing  a  deflec- 
tion 0,  K  being  the  factor  of  the  instrument.  Here  we  may 
wish  to  know  what  is  the  most  advantageous  deflection  0  to  be 
used  to  measure  a  given  current  C,  in  order  that  we  may  con- 
struct the  galvanometer  with  a  suitable  value  of  K. 

To  solve,  we  should  write  an  expression  for  A  in  terms  of 
m,  and  proceed  to  find  from  this  the  value  of  m  which  would 
make  A  a  minimum.  This  would  yield  a  correct  result  pro- 
vided that  this  expression  for  A  did  not  contain  f(m)  as  a  fac- 
tor. If  by  dividing  through  by  f(m),  or  in  any  other  way,  it 
can  be  shown  that  the  expression  for  A  contains/"^),  then  this 
factor  must  be  removed  before  determining  the  minimum. 
For  as  we  wish  to  find  the  best  value  of  m  for  a  given  value  of 


SINGLE   COMPONENT.  1 03 

M  we  must  determine  the  value  of  m  which  will  make  A  a 
minimum  for  that  value  of  M,  i.e.,  we  must  treat  M,  and  there- 
fore/(),  as  a  constant  in  finding  this  minimum.  Now  for  all 
functions  for  which  we  can  write  out  directly  the  expression 
for  A/M,  it  is  evident  that  any  expression  for  A  must  be  divisi- 
ble by  M.  Hence  for  all  such  we  should  save  labor  by  writing 
the  former  at  once  instead  of  writing  out  the  expression  for  A 
and  dividing  by/().  It  is  also  clear  that  if  we  remove  the 
factor  f(  )  from  any  expression  for  A  by  dividing  the  left-hand 
member  by  M  and  the  right-hand  by  f(m),  we  shall  have  left 
an  expression  for  A/M,  and  this  it  is  sometimes  possible  to  do 
even  when  we  cannot  write  this  expression  directly,  as  will  be 
shown  in  the  example  on  the  tangent  galvanometer.  It  is 
important  to  note  that  the  value  of  m  which  we  find  from  these 
expressions  for  A/M  is  the  one  which  will  render  the  fractional 
deviation  a  minimum.  If  on  the  other  hand  we  use  the  ex- 
pression for  A  the  value  of  m  found  is  that  which  renders  the 
deviation  measure  a  minimum.  In  finding  the  minimum,  all 
constant  factors  may  be  omitted,  as  they  do  not  affect  the 
result. 

If  the  function  with  which  we  have  to  deal  is  one  contain- 
ing several  components  of  which  we  are  finding  the  best  value 
of  only  one,  then  the  components  other  than  ml  must,  of  course, 
be  treated  as  constants. 

Briefly  then  the  procedure  to  obtain  the  value  of  m  which 
will  render  A  or  A/M  a  minimum  for  any  value  of  M  is  as 
follows : 

Write,  if  possible,  the  expression  for  A/M  for  the  given 
function.  Otherwise  write  the  expression  for  A,  and  remove 
the  factor  /()  if  it  occurs  by  dividing  by/"().  Remove  all 
constant  factors.  Differentiate  the  resulting  expression  with 
respect  to  m,  equate  to  zero,  and  solve  for  m.  The  criterion 
that  the  second  differential  coefficient  must  be  negative  may 
usually  be  neglected,  inspection  serving  to  determine  whether 
the  value  found  corresponds  to  a  maximum  or  a  minimum. 

There  is  no  best  value  of  m  for  certain  functions  as  follows. 
For  M '=  am  when  8  or  d/m  is  given,  since  in  that  case  A  is  a 


IO4  BEST  MAGNITUDES  OF  COMPONENTS. 

constant  independent  of  m.     For  M  =•  am*  when  6/m  is  given, 

/i          ft  A 

for  j£  =  /— .  which  is  independent  of  m  as  —  is  constant. 

Example  XXV,  page  1 10. 

For  two  variable  components  the  problem  takes  this  form. 
Given  the  form  of  the  function  /"(;«, ,  m^ ,  .  .  . ,  mn)  and  the 
values,  either  numerical  or  relative,  of  tf  or  d/m  for  any  two  of 
the  variables,  e.g.,  for  ml  and  m^ ,  to  find  the  best  ratio  of  those 
variables  and  their  best  numerical  magnitudes.  If  n  >  2,  then 
the  components  other  than  the  two  considered  must  be  treated 
as  constants.  The  following  discussion  applies  only  when  /«, 
and  ;/z,  are  independent  of  each  other.  The  given  values  of 
8  or  d/m  constitute  what  may  be  called  the  precision  condi- 
tions. The  problem  separates  naturally  into  two  parts:  first,  to 
find  the  best  ratio  mjm^\  second,  to  find  the  best  numerical 
values  of  m^  and  m^ ,  to  solve  which  we  must  previously  deter- 
mine the  best  ratio.  It  may  occur  that  only  the  best  ratio  is 
required.  We  will  consider  first  the  finding  of  the  ratio,  be- 
ginning with  the  case  where  #  and  not  8/m  is  given. 

Best  Ratio. — The  best  value  of  mjml  will  be  the  one  which 
will  make  A  a  minimum  for  a  given  value  of  M.  The  proced- 
ure must  therefore  be,  first,  to  obtain  a  suitable  expression 
for  A  or  Ja,  and  then  to  find  by  the  calculus  the  value  of  mjml 
which  will  make  this  a  minimum. 

As  to  what  is  a  suitable  expression  for  A  or  A*  we  may 
readily  see  several  things.  First,  it  must  of  course  be  a  func- 
tion of  dl ,  #„ ,  ml ,  and  m^ .  If  it  is  a  function  of  #,  and  tfa 
alone,  e.g.,  A*  =  d,2  +  <S,3,  it  shows  that  there  is  no  best  value 
for  the  ratio,  for  A  is  determined  independently  of  ml  and  m^. 
Second,  if  it  should  be  found  to  contain  f()  as  a  factor,  that 
factor  must  be  omitted  in  deducing  the  minimum.  For  if  we 
desire  to  find  the  ratio  which  will  make  A  a  minimum  for  any 
given  value  of  J/we  must  treat  M,  and  therefore /(),  as  a  con- 
stant. Now,  if  we  write  the  expression  for  A1  for  any/(),  for 

/4V 

which  we  can  also  write  an  expression  for  \-jffj   by  any  of  our 

formulae,   this   expression   for   A1  must  be   divisible   by  M\ 


TWO  VARIABLE  COMPONENTS.  Io$ 

Hence  it  will  be  simpler  to  write  at  once  the  expression  for 

/  J\2 

\MI  '  w^ere  t^ie  function  is  sucn  that  we  can  do  so,  and  proceed 

to  find  the  ratio  which  will  make  that  expression  a  minimum. 
If  the  expression  for  A*  has  been  written  and  there  is  any  pos- 
sibility that  it  contains  /3( )  as  a  factor,  the  test  for  it  should 
be  made  by  dividing  through  by  it.  If  A/m  can  be  directly  ex- 
pressed, or  if  the  expression  for  A*  proves  to  be  divisible  by/a(  ), 
then  there  is  no  minimum  for^,  but  there  may  be  one  for  A/m. 
Any  constant  factor  of  the  whole  expression  may  be  removed, 

since  it  is  of  no  effect  upon  the  determination  of  the  minimum. 

»  » 

If  instead  of  d.  and  #„  we  have  given  —  and  — ,  there  can 

m^  m^ 

I  A  \Q 
be  no  best  values  for  any  function  for  which  we  can  write  \-jrf] 

directly  in  terms  of  ^Jm^  and  djmv     For  from  the  expression 

\Tf)  =  (/— L)  +\7'*y  >  which   is  the  general  one  for  such 

A 
functions,  it  is  evident  that  -jr=-  is   determined   by  the  given 

d  d 

values  of  — —  and  — -  independently  of  the  ratio  or  values  of 
m^  mt 

ml  and  m^ .  But  with  these  data  the  case  A*  =  d^  -f-  <V  ls 
soluble,  since  from  the  data  we  have  #,  =  const.  X  m^  and 
tf,  =  const.  X  m* ,  which  will  give  us  an  expression  for  A  in 
terms  of  ml  and  m^. 

The  procedure  then  briefly  stated  would  be  : — Write  the 
expression  for  f  (mt  ,  m9 ,  .  .  . ,  mn)  showing  properly  all  the 
measured  components.  From  this,  write  out,  if  possible,  the 


expression   for  [jr^j  .     Otherwise  write  the  expression  for  A* 

and  remove  the  factor  f\  )  if  it  occurs.  Remove  all  constant 
factors.  Find  by  the  calculus  the  ratio  m^/ml  which  will  make 
the  resulting  expression  a  minimum.  The  cases  for  which  there 
is  no  minimum  are  where  f()  =  aml  +  bm^ ,  given  6l  and  tfa ; 

d  d 

and  where  /(  )  =  a-mf  •  m*,  given  — —  and  — — . 

ml          m^ 

To  find  the  minimum,  as  m^  and  m^  are  independent,  it  is 


106  BEST  MAGNITUDES  OF  COMPONENTS. 

necessary  only  to  differentiate  successively  with  respect  to  m^ 
and  mt ,  equate,  and  solve  for  mjm^  For  the  condition  for  a 
minimum  is  that  the  first  differential  coefficients  shall  be 
simultaneously  equal  to  zero.  The  further  conditions  for  dis- 
criminating between  maxima,  minima,  and  points  of  inflection 
need  not  be  considered,  as  inspection  will  show  more  easily 
whether  the  result  obtained  corresponds  to  a  minimum. 

If  ml  is  a  function  of  m^  and  there  are  but  two  components, 
the  solution  cannot  be  made  for  their  best  ratio,  for  their  ratio 
is  fixed  by  the  function.  If,  however,  ml  is  a  function  of  m^ 
and  a  third  component,  then  the  best  value  of  mjml  may  be 
found,  provided  that  the  third  component  is  unrestricted  in 
magnitude.  In  the  solution  for  best  ratio  the  function  must 
be  an  expression  containing  all  the  measured  quantities,  just  as 
for  all  other  precision  problems.  If  ml  is  a  function  of  m^  and 
/#3 ,  as  above  supposed,  so  that  M  =  f(m1 ,  m^ ,  m% ,  .  .  . ,  mn)  and 
ml  =  F(m9 ,  m3),  or  my  =  F(ml ,  ma),  or  m9  =  F(ml ,  m9),  then 
this  function  F( )  is  not  to  be  expressed  in  finding  the  best 
ratios,  unless  it  is  made  use  of  in  computing  M  from  the  com- 
ponents. It  must,  however,  be  employed  in  determining  the 
best  numerical  magnitudes,  if  the  magnitude  conditions  involve 
it.  Illustrations  of  this  will  be  seen  in  the  example  on  moment 
of  inertia. 

Best  Magnitudes.  Two  Components. — To  find  the  best  nu- 
merical values,  it  is  necessary  to  deduce  first  the  best  ratio  by 
the  foregoing  methods. 

As  this  ratio  is  the  best  one  for  any  given  value  of  M,  a 
numerical  value  of  M  must  be  given,  if  M  is  a  function  of  only 
two  components  mv  and  m^ ,  before  those  of  the  two  compo- 
nents can  be  computed.  Thus  having  given  M,  f(ml ,  m^ 
and  mjml ,  the  numerical  values  of  m^  and  m^  can  be  found  by 
direct  substitution. 

Example  XXVI,  page  1 1 1. 

If  the  function  is  of  more  than  two  independent  components, 
but  we  are  dealing  with  only  two  of  them,  ml  and  m^ ,  then  in 
order  to  compute  the  best  numerical  magnitudes  we  must  have 
given,  in  addition  to  the  above,  the  values  of  the  remaining 
components. 


SEVERAL   COMPONENTS. 

If  the  function  is  of  more  than  two  components,  but  m^  and 
m^  are  conditioned  by  a  third  component,  so  that  ml  =  F(m^  mt), 
then  this  function  F()  must  be  given.  In  addition  to  F()  we 
must  have  given  the  value  of  m3  or  such  other  data  as  will  en- 
able us  to  compute  its  value.  It  is  not  necessary  in  this  case 
that  M  should  be  given,  since  F( )  alone  determines  the  best 
ratio,  and  with  m3  the  best  magnitudes,  of  ml  and  m^. 

The  given  value  of  M,  or  of  m^  with  the  function 
m^  —  F(mlt  ;/z2),  mav  be  called  the  magnitude  conditions,  to 
distinguish  them  from  the  given  precision  conditions  under 
which  the  best  ratio  is  determined. 

Example  XXVII,  page  112. 

For  Several  Components.  Best  Ratios. — The  procedure  for 
three  or  more  components  is  the  same  in  character  as  for  two. 
The  best  ratios  are  first  to  be  found.  For  this  the  proper  ex- 
pression for/(^j,  a#a,  .  .  .,  mn)  in  terms  of  the  measured  com- 

MV 

ponents  is  written  out.     From  this  the  expression  for  I —  I    is 

written  out,  if  possible ;  otherwise  that  for  J2  and  the  factor 
/3()  removed  by  division  if  it  exists.  Constant  terms  which  are 
factors  of  the  whole  expression  are  also  removed.  The  remain- 
ing expression  is  then  differentiated  successively  with  respect 
to  m^ ,  m^ ,  and  so  on  for  all  of  the  components  under  considera- 
tion. All  of  these  coefficients  must  be  simultaneously  equal 
to  zero.  Hence  to  find  the  ratio  for  any  pair  of  components, 
we  have  only  to  equate  the  corresponding  coefficients  and  solve 
for  the  ratio.  It  is  convenient  to  find  the  ratio  of  each  to  one 
common  component,  e.g.,  of  mjmlt  m%/mlt  etc.  Inspection 
must  be  resorted  to,  in  order  to  discriminate  between  minima 
and  maxima,  but  this  is  usually  a  very  simple  matter. 

The  cases  for  which  there  is  no  minimum  are  the  same  as 
for  two  variables.  If  some  of  the  components  are  conditioned 
by  being  functions  of  others,  the  procedure  is  the  same  as  for 
two  components,  and  one  free  component  must  exist  in  each 
of  these  functions. 

Exanples  XXVIII,  page  115,  and  XXJX,  page  118. 

Best  Magnitudes. — For  this  part  of  the  solution  the  require- 


IO8  BEST  MAGNITUDES  OF  COMPONENTS. 

ments  and  procedure  are  so  closely  the  same  as  for  two  com- 
ponents that  it  is  not  necessary  to  restate  them.  In  the  sub- 
stitutions it  is  convenient  to  replace  all  the  components  by 
their  equivalents  in  terms  of  the  common  component  and 
solve  for  this  :  then  to  obtain  all  the  others  by  successive  sub- 
stitution in  the  values  for  the  best  ratios. 

Approximate  Solution  by  Equal  Effects. — Where  the 
number  of  components  is  more  than  one  an  approximate  solu- 
tion may  be  used  based  on  the  equations  for  equal  effects.  Its 
advantage  is  greater  simplicity  and  less  labor ;  its  disadvantage 
is  that  it  is  only  approximate,  sometimes  only  roughly  so. 
With  the  same  values  of  d  for  the  components  the  value  of  A 
from  magnitudes  determined  by  the  approximate  solution  may 
be  several  times  as  great  as  by  the  exact  solution,  but  usually 
it  will  not  be  materially  greater.  It  is  convenient  for  com- 
plicated functions.  It  should  be  remembered  that  the  same 
arguments  which  support  the  method  of  equal  effects  in  solving 
for  best  precision  of  components  do  not  hold  in  the  present 
application  of  it,  for  the  question  of  the  labor  involved  does 
not  here  enter. 

Best  Ratio. — The  procedure  is  merely  the  reversal  of  the 
equal  effects  solution  for  best  precision  of  components  given  by 

formula  [28].    Having  given  the  form  of  the  function  f(m1 ,  *#a, 

PI     j> 
.  .  . ,  mM)  and  the  values  of  S1 ,  #, ,  etc.,  or  of  --,  — ,  etc.,  the 

mi  m* 

ratios  of  my/ml ,  m^/ml ,  etc.,  which  will  fulfil  the  conditions  of 
equal  effects,  are  calculated ;  and  it  is  assumed  that  these  will 
make  A  approximately  a  minimum. 

Thus  the  general  equation  for  A  being 


the  general  equations  for  equal  effects  will  be 


APPROXIMATE   SOLUTION  BY  EQUAL  EFFECTS.        1 09 

or,  substituting  the  general  values  of  A^ ,  etc., 


in  which  the  last  term  is  not  needed  to  find  the  ratios  merely, 
but  maybe  useful  in  some  cases  in  finding  the  best  magnitudes. 
Instead  of  this  general  formula  we  may  employ  the  special 
one  for  equal  effects  which  corresponds  to  the  given  function. 

To  make  the  solution  then  we  have  merely  to  substitute 
the  values  of  tf, ,  da ,  etc.,  in  the  suitable  equal  effects  equations, 
and  solve  successively  for  m9/ml ,  m^/ml ,  mjml ,  etc. 

If  the  expression  A*  =  A*  + .  .  .  +  A*  has  a  factor  /'(  ), 
each  term  must  have  that  factor.  And  as  in  the  solution  for 
best  ratios  each  term  is  equated  to  another,  these  common 
factors  cancel.  Therefore  all  cases  are  soluble  directly  from 
the  expression  for  A,  and  it  is  not  necessary  to  remove  the 
factor /2(  ).  A  similar  inspection  shows  that  the  same  solution 
will  be  arrived  at  whether  we  start  from  the  expression  for  A* 

I A  \2 
or  for  f  -JT-J.  \   in  a  case  where  the  latter  is  applicable. 

Best  Magnitudes. — These   are   determined  from   the   best 
ratios  just  as  by  the  exact  method. 
Example  XXX,  page  118. 


110  BEST  MAGNITUDES  OF  COMPONENTS. 


EXAMPLES. 

Example  XXV.  —  Best  Magnitudes.  One  Component.  — 
Given  a  tangent  galvanometer  read  by  an  index  moving  over 
a  circle  graduated  in  equal  parts.  Let  0  be  any  reading,  and 
K  the  galvanometer  factor.  Then  the  current  is 


[89] 


The  deviation  measure,  #0,  of  a  single  reading  will  be  the 
same  at  all  parts  of  the  scale.  What  would  be  the  best  deflec- 
tion to  use  ?  This  problem  might  arise  from  either  of  the  two 
following  questions.  For  a  given  current  how  should  K  be 
proportioned  in  order  that  the  value  of  AC  for  the  given  value 
of  #0  should  be  a  minimum  ?  Or  if  C  were  variable  at  will 
and  K  were  given,  what  value  of  0  would  give  C  with  the 
greatest  fractional  precision  as  far  as  £0  affects  it  ? 

The  two  problems  have  the  same  solution  ;  for,  in  the  first, 
when  dCiss.  minimum  AC/C  will  also  be  so,  as  C  is  a  constant. 
K  may  be  omitted,  although  it  is  a  component  which  must  be 
measured,  for  no  limitation  is  assigned  to  it  in  the  statement 
of  the  problem,  and  we  are  therefore  to  assume  it  as  determi- 
nable  with  any  desired  precision  or  with  equal  precision  what- 
soever its  value. 


dml 


Dividing  by  /(  )  to  test  whether  it  is  a  factor,  we  have 


.  > 

K-tan<f>  sin  0-cos  0  sin  20 


EXAMPLES.  1 1 1 

which  shows  that  the  factor  /( )  exists  and  leaves  us,  as  the 
expression  to  be  made  a  minimum,  omitting  the  constant 
'factor  2, 

i 
sin  20* 

It  is  easy  to  see  by  inspection  that  this  is  a  minimum  for 
0  =  45° ;  but  proceeding  with  the  general  method,  we  have 

d         I  d  ,  cos  20 


d(f>  sin  20       dcj)  sin2  20 

.-.  cos  20  =  o,     and     0  =  45°, 

as  the  best  value  of  0,  answering  either  requirement. 

Example  XXVI. — Best  Magnitude.  Two  Components. — 
The  rate  h  of  production  of  heat  in  a  conductor  is  to  be  de- 
termined by  measuring  the  current  c  and  the  voltage  v.  The 
instruments  available  can  measure  the  current  with  a  precision 
of  dc=  0.05  ampere,  and  the  voltage  to  dv  =  0.05  volt.  What 
are  the  best  values  of  c  and  v  for  a  rate  of  heating  of  25 
calories  per  second.  For  grm.,  deg.  C.,  true  volts  and  ohms 
we  have,  at  a  temperature  of  15°  C, 

h  —  0.2387  vc. [9O] 

The  conditions  are  dv  =  0.05,  dc  =  0.05,  h  =  25. 

There  are  only  two  variable  components,  m^  =  v,  mz  =  c : 
and  both  are  to  be  discussed.  As  the  resistance  of  the  conduc- 
tor is  not  specified,  they  are  independent.  Expression  [90]  for 
/(  )  contains  all  the  measured  quantities,  and  is  such  that  we  can 

write  the  expression  for  f-^J  ,  viz., 


A\      tdv\*  ,    (dcy     6*v  .   PC 

M)  ==  lir;  +  (-)  =  -jr  •-•?•• 


Then 

*.(—^  -  —  — 

dv\M)    -~2  v*  '          dc\M 


112  BEST  MAGNITUDES  OF  COMPONENTS. 

Equating  to  zero  simultaneously  and  solving  we  have 


.'.  —  =  I.  is  the  best  ratio. 
v 


To  find  the  best  numerical  magnitudes  we  may  substitute 
this  ratio  and  the  value  of  h  in  the  original  expression  and 
solve.  Thus 

h  =  0.24  v  X  v  =  0.24  v*  ; 


c  =  v  =  10.  amperes. 

This  problem  is  purposely  made  numerically  simple  so  that  in- 
spection may  readily  show  the  results  to  be  correct. 

Example  XXVII.  —  Best  Magnitudes.  Two  out  of  Three 
Components.  —  A  bar  is  to  be  constructed  whose  moment  of 
inertia  may  be  computed  from  its  measured  mass  m  and  its 
linear  dimensions.  It  is  to  be  a  right  circular  cylinder  of  height 
h  and  diameter  </,  and  is  to  swing  about  a  transverse  central 
diameter.  Both  h  and  d  are  to  be  measured  with  the  same 
micrometer  screw,  but  owing  to  the  uncertainty  from  rounded 
edges  and  other  imperfections  of  the  ends  6h  =  ^dd.  The  mass 
m  is  to  be  found  by  weighing,  and  can  be  determined  so  closely 
that  8m/  m  is  negligible.  What  is  the  best  ratio  of  d/h  ? 

Here  there  are  three  measured  components,  but  one  of 
them,  m,  is  omitted  from  the  discussion  because  its  6m/m 
is  negligible.  This,  moreover,  is  a  function  of  the  other  two 
components,  so  that  unless  it  or  one  of  the  others  could  be 
omitted,  the  problem  could  not  be  solved.  The  expression  for 
the  moment  of  inertia  is 


EXAMPLES.  113 

/  A  \2 
The  function  f(  )  is  such  that  we  cannot  write  out  f-r     for 


it  by  any  of  the  special  formulae.    Applying  the  general  formula, 
we  have 


df()  _  mh      df()  _  md 
~ 


This  is  not  divisible  by  the  expression  f(  ),  hence  we  cannot 
take  out/(  )  as  a  factor.  Omitting  then  the  common  constant 
factor  m*/4,  we  have  as  the  expression  to  be  made  a  minimum 


._     . 

9  16 

which  will  be  denoted  by  [  ].     Then 


dh  da 


_  ___ 

~:  y  P  "  ~c7  " 

The  best  ratio  then  is  d/h  =  28,  that  is,  the  cylinder  should 
be  a  very  short  one,  a  disc  rather  than  a  long  cylinder.  This 
result  is  rather  striking,  as  the  first  thought  might  be  that  6  A 
being  greater  than  3d,  h  should  be  made  greater  than  d,  so  that 
the  fractional  accuracy  of  h  should  be  increased.  Further 
consideration  and  inspection  of  the  formula  for  I  will,  however, 
show  that  the  result  arrived  at  is  rational  under  the  conditions 
of  dh  =  4.dd.  Moreover,  if  the  value  of  ^a  be  computed  for 
this  ratio,  and  then  for  another  ratio  quite  different,  but  which 
makes  the  value  of  /  the  same,  the  value  of  A  will  be  found  to 
be  greater  in  the  second  case.  A  test  of  that  sort  will  also 


114  BEST  MAGNITUDES  OF  COMPONENTS. 

show  another  thing  which  has  elsewhere  been  noted,  namely, 
that  the  value  of  A  will  change  but  little  for  considerable 
changes  in  the  ratio.  This  is,  however,  dependent  on  the  form 


To  find  the  best  numerical  magnitudes  for  h  and  d,  we  may 
have  the  necessary  magnitude  conditions  given  in  several  ways  ; 
two  examples  will  be  taken. 

First.  The  most  usual  form  of  the  problem  would  be  this. 
To  find  the  best  values  of  h  and  d  for  a  bar  of  stated  moment 
of  inertia  and  material  ;  for  example,  moment  of  inertia  to  be 
1500  c.  g.  s.,  and  the  bar  to  be  of  brass.  Since  m  =  ^nd*hp, 
where  p  is  the  density,  we  must  know  p  in  order  to  find  what 
m  would  be.  Suppose  for  the  brass  p  =  8.5.  The  magnitude 
conditions  would  then  be  /—  1500  and  p  =  8.5  ;  and  there 
would  be  the  equation  m  =  ^nd^hp  conditioning  m,  h,  and  d. 

To  find  the  best  value  of  d  we  may  substitute  the  magni- 
tude conditions  and  the  value  h  =  d/2%  in  the  full  expression 
for  /,  viz., 


.-.  i5oo  =  -  X  3-i  X  d*  X  -x-  X  8, 
4  zo 

.-.  d*  =  100  ooo.     d  =  10.  cm. ;    h  =  10/28  =  0.36 


cm. 


Second.  It  might  be  required  to  construct  the  bar  of 
brass  and  with  a  stated  mass,  e.g.,  m  =  63.  gms.  The  mag- 
nitude conditions  would  then  be  m  =  63.  gms.,  p  =  8.5,  and 
there  would  be  the  same  condition  equation  m  =  \nd*hp. 

Then  d  and  h  must  fulfil  these  conditions  and  have  the 
ratio  d/h  =  28,  simultaneously.  We  must  therefore  eliminate 
d  and  h  successively  between 

63  =  -TT^  x  8.5,     and     j-  =  28. 
4  n 

.-.  63  =  £  X  3-i  X  28'A2  X  h  X  8.5. 

h*  =  0.0122,     h  =  0.35  cm.,     d  =  2%h  =  9.8  cm. 


EXAMPLES.  115 

These  results  of  course  agree  with  those  of  the  first  case  within 
the  limits  of  error  of  two-place  computations,  the  data  being 
equivalent. 

Example  XXVIII.  —  Best  Magnitudes.  Several  Components. 
—  The  modulus  of  elasticity  E  of  an  unplaned  wooden  beam 
IO  ft.  long  is  to  be  determined  by  measuring  the  weight  W 
at  its  centre  necessary  to  produce  a  transverse  central  deflec- 
tion v  when  supported  at  the  ends,  and  by  measuring  the  mean 
breadth  b,  depth  h,  and  the  length  /.  The  value  of  E  is  known 
in  advance  to  be  about  1.3  X  io6  Ibs.  per  sq.  in.,  and  exami- 
nation of  the  beam  and  measuring  apparatus  shows  that 
the  precision  attainable  will  be  61  =  0.5  in.,  db  =  dh  =  0.05  in., 


dv  =  0.002  in.,  -f~-  =  o.ooi.     Desired  the  best  magnitudes  of 

the  components. 

The  expression  for  E  is 


The  components  are  connected  by  no  equation  of  condition 
among  themselves,  so  that  the  best  ratios  might  be  found  for 
all  of  them  if  the  precision  conditions  permitted,  were  it  not 
for  the  fact  that  the  length  /  is  specified.  But  we  can  write 

/  J\3 
the  expression  for  \jr?\    directly  by  [57],  viz., 


Now  as  -™-  is  a  constant  by  the  conditions,  its  effect  on  M 

will  be  the  same  whatever  the  value  of  W',  hence  there  will  be 
no  best  magnitude  for  W,  and  it  may  be  omitted  from  the  dis- 
cussion. This  fact  enables  us  to  introduce  the  condition 


n6 


BEST  MAGNITUDES  OF  COMPONENTS. 


l—i2o  in.    Proceeding  then  with  the  others  to  find  the  best 
ratios,  we  have 


Hence 


_I8_= 


__ 
~  °3' 


db 


' 


rf(_) 
dh 


v/  j  o \/_  2 


6*b        b*       I     &b 
2_       ?  =  _  ._ 


.-.  -  =  o.ia 


-I8r  =  -i8-l    -  =  — =  0.oia 


/.  -7  =  O.22. 


7-3-  =  -  '  TF7  =  O.OOOOO  l8.     .'.  -7  =  O.OI2. 

/         9     o  /  / 

To  find  the  best  magnitudes,  /=  120.  in. 
.•.  b  =  o.io/   =  12.  in. ; 
h  =  0.22  /   =  26.  in. ; 
v  =  o.oi2/=  1.4  in. 

Note  that  as  the  components  are  all  independent,  and  as  W 
may  have  any  value  as  far  as  the  conditions  show,  we  should 
be  obliged  to  assign  a  value  to  some  component,  if  not  to  /. 

To  see  whether  these  best  magnitudes  are  practicable  we 
should  compute  W  to  see  whether  it  came  within  the  limits  of 
the  testing-machine.  Transposing  and  substituting  gives 


4bh*vE       4  X  12  X  26*  X  1.4  X  1.3  X  io6 


r 


I203 


=  i.i  X  io6. 


But  i  100  ooo  Ibs.  would  be  beyond  the  limits  of  most  ma- 
chines, so  that  we  should  be  obliged  to  use  some  smaller  values 


EXAMPLES.  117 

for  b,  h,  and  z>,  and  should  wish  to  maintain  the  same  ratios. 
Let  us  then  limit  the  load  to  W  =  100  ooo.  Ibs.  We  must 
therefore  have 

Wr  io5  X  i2o3 

***=-^  =  4Xi.3X.o—  3-3XK?; 

also 

b       o.io  v       0.012 


Substituting  the  latter  gives 

0.45  X  0.05  s^5  =  3.3  X  io4.    .%  0.025^  =  3.3  X  io4. 
.-.  k  =  17.  in.  ; 

b—    0.45  X  17.  =  7-7  in-; 
v  =  0.055  X  17.  =  0.94  in. 

This  result  may  be  checked  by  substituting  the  values  in  the 
expression  for  E,  which  should  yield  1.3  X  io6  within  one  or 
two  units  in  the  second  place. 

To  see  how  much  difference  there  would  be  in  the  precision 
of  E  under  the  two  cases  we  may  compute  A/M  for  each. 

For  the  first  values, 


=  o.oo  ooo  i  +  °'O°  OI4  +  o-oo  ooi  ^  +  o.oo  003  6  -f-  o.oo  ooo  3 
=  000  020. 
A 


For  the  second  values, 


°^2  ,      /0-°5^a  ,   ^0.002  \8 

—  ^1  4-o(  —  -}  +  -  1 

7.7/     ry\i7.  /     ~\  1.4  / 


ISF]  =0.001+0 
M)         *  i2o 

=  o.oo  ooo  i  +  o.oo  014  +  0.00004  2  +  o.oo  008  i  +  0.00  ooo  3 
=  o.oo  026. 

=  °-016- 


Il8  BEST  MAGNITUDES  OF  COMPONENTS. 

The  close  agreement  of  these  two  values  shows  that  the  second 
set  of  magnitudes  is  sensibly  as  good  as  the  first. 

Example  XXIX. — Best  Magnitudes.  Conditioned  Compo- 
nents.— The  specific  resistance  per  metre-gramme  of  a  sample 
of  copper  wire  is  to  be  determined  by  measuring  the  resistance, 
R,  and  mass,  m,  of  a  measured  length  /  of  the  wire.  Given 
3R  =  o.ooi  ohm,  dm  =  o.ooi  grm.,  dl  —  0.3  mm. 

In  this  form  the  problem  is  insoluble.  For  as  the  material 
of  the  wire  is  stated,  the  mass  is  a  function  of  the  length, 
diameter,  and  specific  gravity,  and  the  resistance  is  also  a  func- 
tion of  the  length,  diameter,  and  resistance  per  unit  volume, 
and  as  both  specific  gravity  and  resistance  per  unit  volume  are 
constants,  R,  m,  and  /  are  conditioned  quantities.  This  may 
be  expressed  in  another  way  by  saying  that  as  the  material  is 
fixed,  then  for  any  given  value  of  ;«//,  that  is,  of  mass  per  unit 
length,  there  is  a  fixed  value  of  R/l,  that  is,  of  resistance  per 
unit  length.  We  are  therefore  not  at  liberty  to  assign  ratios 
R  :  m  :  I  on  the  basis  of  the  precision  conditions.  It  could  be 
solved  for  R  :  I  under  the  condition  dm  negligible. 

The  above  form  of  the  precision  conditions  is  not,  however, 
the  one  which  would  ordinarily  arise  in  practice.  We  should 
usually  have  given  dR/R,  and  #///,  that  is,  R  and  /  would 
each  be  measurable  with  a  constant  fractional  precision.  And 
6m/m  would  be  usually  negligible,  measurements  by  the  balance 
available  being  generally  far  more  precise  than  the  measure- 
ments of  either  R  or  /.  In  this  case  there  would  also  be  no 
best  ratios,  since  4 /Mis  a  constant,  being  fixed  by  the  precision 
conditions  independently  of  the  values  of  the  components. 
The  best  values  would  be  determined  by  other  limitations  of 
the  apparatus. 

Example  XXX. — Best  Magnitudes.  Equal  Effects. — In  il- 
lustration of  this  method  the  preceding  example  on  the  deter- 
mination rf  E  for  a  wooden  beam  will  be  taken.  Applying 
the  method  we  have 

SW _     d/        _dtf  <^  _to, 

W  ~3  /  ""     b  -  ~3  h    ~   v' 


EXAMPLES.  1  19 

SW 
Then  for  -777  constant,  we  have  for  the  best  ratios 

61  6b         b  i     6b 

*T     •-*->    ~r~  -rov  =   - 

dl  3k      h  3h 

3T  =  ~3T'    7  =  "  tf7=-ai 

61  dv         v  I     dv 

3T:      V     7  =  -J'd/  =  - 

The  negative  signs  indicate  merely  that  an  increase  in  that 
component  causes  a  decrease,  or  a  negative  error,  in  the  result. 
For  best  magnitudes  with  /=  120  in.  we  then  have 

b  =  0.033  I  —  4-°  m-  J 
h  =  o.io  /=  12.  in.  ; 
v  =  0.0013  /=  0.16  in. 

These  dimensions  would  require  a  load 


which  is  small  for  the  capacity  of  the  machine.  One  objection 
to  the  dimensions  would  be  that  they  are  too  small  to  corre- 
spond to  commercial  sizes.  They  would  be  modified  as  in  the 
former  example. 


SOLUTIONS  OF   ILLUSTRATIVE 
PROBLEMS. 


Example  XXXI.  —  Problem.  —  A  voltmeter  is  to  be  calibrated 
by  the  Poggendorff  method,  using  a  Carhart-Clark  cell.  The 
calibration  at  no  volts  is  desired  to  0.2  volt,  of  which  error 
one  half  is  allowable  in  the  cell  measurement,  the  other  o.i 
volt  being  assigned  to  the  voltmeter.  The  arrangement  used 
is  to  connect  the  voltmeter  in  series  with  a  battery  of  over  1  10 
volts,  a  controlling  water  rheostat,  and  an  accurate  adjustable 
resistance  r,  the  Clark  cell  being  connected  around  this  resist- 
ance. The  water  rheostat  is  adjusted  until  the  voltmeter 
reads  1  10  volts,  and  the  resistance  r  until  on  closing  a  key  in 
the  cell  circuit  no  deflection  occurs  on  a  sensitive  galvanome- 
ter in  that  circuit,  an  approximate  adjustment  being  effected 
at  first  by  a  preliminary  computation,  in  order  to  avoid  injury 
to  the  cell. 

Solution.  —  Let  R  denote  the  voltmeter  resistance;  r,  the 
adjustable  resistance  ;  /,  the  observed  temperature  of  the  cell 
E  =  1.438  legal  volts  (1884),  the  voltage  of  the  cell  at  15°  C.  ; 
a  =  —0.00038,  the  temperature  coefficient  of  the  cell;  then 
the  voltage  at  the  terminals  of  the  voltmeter  will  be 


[94] 


By  computation,  if  V=  no,  R  =  17000,  and  /  =  15°,  we  find 
r  =  220  ohms  approximately,  which  would  be  about  the  amount 
which  we  should  require  to  obtain  the  balance. 

120 


EXAMPLE  XXXI.  121 

To  find  the  accuracy  requisite  in  the  5  components  Ey  a, 
t,  R,  and  r,  we  apply  the  general  formula  [38].  We  will  assume 
/  =  20°  as  a  typical  value. 


_R_  _  17000.  _ 

dE~~~-  r  ~^o7       8°' 

i  rr  r> 

•^-=  -E-(t-i$)  =  -  1.4  X  8ox(20-  15)  =  -560.; 

i  -TT  rj 

-j-  —  —  Ea—  —  —  1.4  X  80  X  0.00038  =  —  0.042  ; 

dV      E  i 

-,,-  =  I.4/220  =  _. 

dV  R  8o  i 


J  V  —  o.io  volts.     4V/  Vn  =  o.io/  ^5  =  0.045  volts. 

<y£"  =  0.045/80.  =  o.oo  056^.  Attainable.  There  is  probably 
a  constant  error  of  more  than  half  of  this  amount 
in  the  absolute  value  of  the  ohm. 

da  =  o.O45/(—  560)  =  —  0.00008  o.     Easily  made  negligible. 
dt  =  o.Q45/(-  °-°42)  =  -  i°-i. 
SR  =  0.045  X  1  60    =       7.2  ohms      =  0.04  %.  )     As  only  the 

dr  =  0.045  X  (—  2)  =  —  0.090  ohms  =  0.04  %.  )  ratio  of  R  :  r 
is  required,  the  necessary  precision  may  be  reached; 
but  with  German-silver  coils  a  variation  of  i°  in 
temperature  would  correspond  to  this  amount. 
Thus  the  precision  measure  attainable  would  be  about 


j^  =  o.ooo  68  —  0.068  per  cent, 

which  is  slightly  better  than  the  o.i  per  cent  called  for,  which 
may  therefore  presumably  be  attained. 


122  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

Example  XXXII.— Efficiency  of  Electric  Generator  or 
Motor  by  Stray-power  Method. 

Explanation  of  Method. — If  in  a  generator  we  let  L  denote 
the  total  power  losses  in  the  machine,  whether  electrical  or 
mechanical,  and  if  we  let  the  mechanical  power  applied  to  the 
machine,  i.e.,  the  input,  be  denoted  by  /,  and  the  electrical 
power  available  in  the  circuit  outside  of  the  machine,  in  other 
words  the  electrical  output,  by  (9,  then  the  "  commercial  effi- 
ciency" of  the  generator  is 

E  =  -ft      ."V [95] 

and  the  total  power  losses  are 

L=I-O,    ...     .  -  «    ...     \96] 

all   power    being,   of    course,   expressed   in    the   same    unit. 
Whence 

7=0+ A [97] 

and 

O 


[98] 


From  this  last  expression  it  is  obvious  that  if  we  have  any 
means  of  measuring  the  losses  L  we  may  ascertain  the  effi- 
qiency  from  measurements  of  L  and  O  alone  without  measur- 
ing /.  Several  methods  exist  by  which  this  can  be  done,  some 
of  which  measure  L  by  electrical,  others  by  mechanical  methods 
and  still  others  by  a  combination  of  the  two. 

The  above  formula  for  E  for  generators  must,  of  course,  be 
slightly  modified  to  be  applicable  to  motors.  Let  i  be  the 
electrical  input  or  electrical  power  supplied  to  the  motor,  and 
o  its  mechanical  output.  Then  its  commercial  efficiency  is 


[99] 


EXAMPLE  XXXII.'  123 

Then,  as  before,  if  /  denotes  the  total  losses  in  the  machine, 
we  have 


and 


whence  by  measuring  i  and  /  we  may  calculate  e.  As  for  the 
generator  so  for  the  motor  there  are  several  methods  by  which 
/  may  be  measured  electrically  or  mechanically. 

Of  the  electrical  methods  for  generators  and  motors  the 
simplest  is  the  •'  Stray-power"  method.  This  is  applicable  to 
a  wide  variety  of  machines,  and  forms  perhaps  the  best  avail- 
able method  for  general  technical  testing.  In  common  with 
all  methods  which  measure  the  losses,  it  possesses  an  important 
advantage  over  those  methods  which  measure  both  o  and  i. 
For  inspection  of  the  expression  for  e  will  show  that  an  error 
of  i  per  cent  in  the  losses  corresponds  only  to  about  o.i  per 
cent  in  e  since  /  is  only  about  one  tenth  of  t,  and  a  similar 
statement  is  true  for  a  generator.  The  discussion  will  also 
show  that  the  number  of  components  to  be  measured  becomes 
so  small  that  this  fact  in  combination  with  the  above  enables 
the  method  to  give  a  considerably  greater  precision  in  the  effi- 
ciency than  is  required  for  any  of  the  components.  Briefly 
explained  the  method  is  as  follows: 

The  losses  of  power  in  a  dynamo  (either  motor  or  gener- 
ator) may  be  divided  into,  1st,  the  loss,  A,  due  to  the  armature 
resistance  ;  2d,  that,  F,  due  to  the  field  resistance  ;  3d,  the  sum 
total  of  the  losses  due  to  all  other  sources,  viz.,  Foucault  cur- 
rents, hysteresis,  bearing  friction,  air  resistance,  etc.  The  third 
group  has  been  termed  the  "  stray  power,"  and  will  be  denoted 
by  SP.  For  a  stated  condition  of  running,  that  is,'at  a  specified 
voltage,  current,  and  speed,  each  of  these  losses  has  a  fixed 
value  provided  that  the  condition  of  steady  temperature  of  the 
machine  has  been  reached. 


124  SOLUTION'S  OF  ILLUSTRATIVE  PROBLEMS. 

The  first  two  losses,  A  and  F,  can  be  calculated  if  the  fol- 
lowing quantities  are  known,  viz., 

ca  =  current  in  armature, 
cf  =          "       "  field  coils ; 
ra  =  resistance  of  armature ; 
r,  —  "          "  field  coils, 

or  vf  -—  voltage  between  field  terminals. 

For 

A  =  c?ra , 

(Vf\*  V* 

F=c}rf,     or     F  =  (-)  r,  = -, 

where  either  cf  or  vf  may  be  used  as  is  most  convenient  in  any 
given  case. 

In  what  follows  let  us  for  simplicity  suppose  that  we  have 
a  simple  shunt-wound  motor. 

The  stray  power  cannot  be  directly  measured,  but  must  be 
indirectly  determined  by  difference.  We  have,  of  course, 

L  =  A  +  F+SP,     ,     .     ;   ';.    .     [1O2] 

Thus  if  L,  A,  and  F  are  measured  under  any  given  condi- 
tion, SP  for  that  condition  can  then  be  deduced. 

The  stray-power  method  rests  upon  this  latter  fact.  It  also 
involves  the  facts  that  the  SP  does  not  change  widely  between 
full  load  and  no  load  on  most  types  of  machine,  that  it  is  of 
sensibly  the  same  amount  for  the  same  machine  whether  acting 
as  a  generator  or  motor,  and  that  its  change  with  slightly  differ- 
ent speeds  of  the  machine  may  be  allowed  for  approximately. 
The  procedure  consists  in  running  the  motor  with  no  load 
under  as  nearly  as  possible  its  rated  voltage  and  speed,  and 
measuring  the  actual  voltage  v,  current  c,  and  speed  s.  The 
armature  resistance  ra,  the  field  resistance  rf,  and  the  current 
cf  or  the  voltage  vf  between  the  field  terminals,  for  the  same 
conditions  must  also  be  measured  or  otherwise  ascertained, 
except  as  shown  below.  Then  the  total  loss  of  power  under 
this  condition  is 

/=», 


STRAY-POWER  METHOD.  12$ 

and  the  stray  power  is 

sp  =  l-(a+f)  =  cv-  efr.  -  cfr,  .     .     [JOS] 

Let  SP  denote  the  stray  power  of  the  machine  under  any 
specified  load  at  rated  speed  »£  and  voltage  V.  Then  it  is  as- 
sumed, first,  that  SP  =  sp  sensibly  if  5  =  s,  and,  second,  that  if 
s  differs  from  5,  then 

SP=-.sj> [104] 

s 

Both  assumptions  are  fairly  well  supported  by  the  com- 
parison  of  results  of  experimental  tests  of  the  same  machine 
by  different  methods,  but  neither  is  exact,  nor  is  either  reliable 
for  machines  of  inferior  design.  The  assumptions  may  proba- 
bly be  regarded  as  introducing  an  error  of  less  than  one  half 
of  one  per  cent  into  E  under  ordinary  conditions  of  testing. 
Hence  the  efficiency  of  the  given  machine  under  the  specified 
load  may  now  be  calculated.  Let  V  be  the  rated  voltage* 
C  the  current  corresponding  to  the  specified  load,  and  5  the 
rated  speed  ;  then  we  have 

l=A         +F       +SP 

I  =  Ca*Ra  +  CfRf  +  -(cv  —  ca*ra  —  c/rf) ; 

*=  CV-, 

--•-       -cv\  f    s™  -*•*•  *frd\ 

Here  the  capitals  denote  the  quantities  for  the  specified  load, 
and  the  small  letters  for  the  measurement  with  no  load,  i.e., 
when  the  stray  power  is  being  measured.  The  quantities  C, 
V,  and  5  are  not  measured,  they  are  merely  specified  amounts. 
The  quantities  Ra  and  Rf  must  be  determined  by  measurement 
or  otherwise  for  the  specified  condition,  and  likewise  Cf,  except 
as  shown  below.  There  are  therefore  at  most  nine  quantities, 
viz.,  Ra,  Rf,  s,  c,  v,  ca,  ra,  cf,  rf,  to  be  measured.  But  the 
precision  discussion  will  show  that  this  number  can,  in  prac- 
tice, be  considerably  reduced. 


126  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

The  foregoing  formula  applies  to  a  motor.  A  generator 
would  be  treated  in  precisely  the  same  way ;  that  is,  it  would 
be  run  as  a  motor  under  no  load  and  c,  v,  etc.,  measured. 
Then  to  obtain  an  expression  for  its  efficiency  as  a  generator 
at  any  specified  rate  of  output  C  and  V  .ve  should  have  merely 
to  substitute  the  values  of  c,  v,  C,  V,  etc.,  in  the  expression  [98] 
for  E.  For  the  purposes  of  the  precision  discussion,  however, 
we  should  not  employ  that  expression,  but  should  simplify  it 
thus: 

approx.  =  I  —  —  approx.    [106] 


This  expression  could  not  properly  be  used  to  compute  E, 
but  is  exact  enough  for  the  precision  discussion.  Thus  we 
should  have  for  the  generator 

£=  i  ~=  i 


j  C.'X.+CyJt^cv-  c.'r.-c/r,)  j  ,     [JOT] 


which  is  identical  in  form  with  that  for  the  motor. 

In  either  case  the  input  or  output  corresponding  to  CV 
may  be  anything  we  choose,  e.g.,  half  load,  three-quarters  load, 
full  load,  etc.  This  will  be  more  fully  perceived  in  the  example. 

Problem.  —  A  shunt-wound  motor  is  to  be  tested  for  com- 
mercial efficiency  e  by  the  Stray-power  Method.  It  is  rated 
at  220  volts,  40  amperes,  and  1200  revolutions  per  minute,  and 
is  stated  by  the  maker  to  have  an  armature  resistance  of  0.14 
ohm  and  a  field  resistance  of  130  ohms.  The  precision  desired 
in  the  value  of  e  for  full  rated  load  is  Ae/e  =  0.25  per  cent. 
Required  to  find  by  the  precision  discussion  : 

(a)  Precision  necessary  in  the  measured  components. 

(b)  Whether  any  of  the  components  can  be  wholly  omitted. 

(c)  How  closely  to  their  normal  running  temperature  the 
field  coils  and  the  armature  must  be  when  measured. 

(d)  Whether  any  of  the  results  of  (a)  could  be  applied  to 
other  motors  ;  and  if  so,  under  what  conditions. 


STRAY-POWER  METHOD.  12/ 

Solution. — For  the  solution  we  require  approximate  values 
of  c,  v,  ra,  rf,  Ra,  and  Rf.  The  better  way  would  be  to  make 
a  preliminary  run  and  measure  these  quantities  roughly.  But 
it  is  usually  more  convenient  to  make  the  discussion  in  advance 
of  any  trial.  We  may  do  so  in  this  case  as  follows :  Assume 
that  ra  —  Ra  =  0.14  ohms,  also  that  rf=.Rf=  130.  ohms,  the 
values  stated  by  the  maker.  Both  of  these  assumptions  will 
be  proved  to  be  close  enough  by  the  results  of  the  discussion. 
We  are  obliged  further  to  assume  a  value  of  e  in  order  to 
deduce  a  value  for  c.  This  we  can  usually  also  do  closely 
enough  for  the  preliminary  discussion  from  inspection  of  the 
machine.  Suppose  that  in  the  present  case  we  estimate  the 
efficiency  to  be  about  88.  per  cent  at  full  load.  Then  if 
i>  =  220.,  c  must  be  0.12  X  40.  =  4.8  amp.,  since  100  —  88 
—  12  per  cent  of  the  power  and  therefore  of  the  current 
applied  under  normal  voltage. 

The  expression  above  deduced  for  e  must  be  slightly  modi- 
fied to  meet  this  case,  for  ca  and  cf  are  not  here  measured,  but 

v  v 

cf  =  — ,     and    ca  =  c  -  —. 


The  expression  for  e,  then,  containing  all  the  components  to 
be  measured  properly  expressed  for  the  precision  discussion  is 


The  components  to  be  measured  in  the  test  are  thus  seven, 
viz.,  Ra  ,  Rf,  s,  c,  v,  ra  ,  and  rf,  so  that  n  =  7.  We  have  there- 
fore to  apply  the  general  formula  [37]  to  these.  The  following 
simplification  may  be  made.  As  s  may  easily  be  made  within 
a  few  per  cent  of  5  in  the  run  we  may  regard  S/s  as  unity  in 
all  differentiations  except  that  with  respect  to  s.  The  values 
of  C  and  Fare  not  measured  values,  but  simply  stated  to  define 
the  condition  at  which  the  efficiency  is  to  be  computed.  Not 
being  measured  they  are  not  subject  to  errors  of  measurement, 


128 


SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 


and  are  therefore  not  to  be  differentiated.     Proceeding  then 
with  the  differentiation  and  substituting  numerical  values  gives 


—  -    —\ic   ^  \ 

dRa~     ~  CV\\L  ^  Rfl  ) 


Rf  R; 


io 


de_ 
ds 

^L 
~fa 

de_ 
de_ 
de 


v\r        2v 


~  7   \ 


d€  -  -  —  |        2\c  -  - 
dr<  CV\          V       Tf 


6.3  X  io3' 

i 
9.1  X  ioa' 

3.0  X~fo3' 


To  find  the  numerical  values  of  dRa  ,  etc.,  by  [37],  we  must 
have  Ae.  Now  ^/V  =  0.0025  and  ^  =  0.88;  .'.^=0.0025 
X  0.88  =  0.0022.  Hence  Ae/  Vn  =  O.OO22/  V?  =  0.0022/2.7 
=  0.00081  .'. 


(a) 


=  —  8.  X  io-4  x  6.3 


8s     = 

dc     = 
dv    = 


X  1.5  X  io4  =  -f  12. 
X  4-0  X  ro1  =  -    0.032 
X  6.3  X  io8  =  -    5.0 
X  9.1  X  io2  =  +   0.73 
X  3-0  X  io3  =  -    2.4 


=  —    0.0050  ohms       =  —  3.6  % 
=  4-    2.5        ohms       :=-j-i.9# 
r.  p.  m.   =-j-  i.o  £ 
amperes  =  —  0.67  % 
volts       =  —  2.3  % 
ohms       =  -|-  500.  % 
ohms       =  —  1.8  % 


(&)  (c)  Inspecting  these  values  the  following  points  may  be 
noted,  (i)  The  armature  resistance,  ra,  in  the  run  under  no 
load  is  entirely  negligible.  A  single  measurement,  viz.,  Ra , 
with  the  armature  at  the  temperature  of  full  load  is  all  that  is 
needed  ;  hence  we  might  use  n  =  6  instead  of  n  =  7.  (2)  The 


STRAY-POWER  METHOD.  I2g 

armature  coils  being  of  copper  will  change  in  resistance  by  0.4 
per  cent  per  degree  centigrade.  The  maximum  allowable 
change  is  3.6  per  cent,  which  corresponds  to  3.6/0.4  =  9.°  C. 
Some  care  is  therefore  essential  that  the  armature  is  fully 
warmed  up  to  its  normal  state  when  Ra  is  measured.  (3)  The 
values  of  6Rf  and  drf  are  nearly  equal  and  of  opposite  sign, 
and  obviously  Rf  =  rf  very  nearly.  Hence  if  the  same  numer- 
ical value  be  used  for  both,  any  error  in  that  value  will  be 
nearly  eliminated,  so  nearly  in  fact  that  a  large  error  will  be 
admissible.  We  may  see  how  large  by  substituting  rf  for  Rf, 
and  then  differentiating  e  with  respect  to  rf.  This  gives 


drf 


,    f  I  V 

+L  -  T  -  7 


.-.  6'rf  =  —  8  X  icr4  X  6.8  X  io4  =  —  54.  ohms  =  —  42.  %. 

This  clearly  shows  that  one  rough  measurement  of  the  field 
resistance  is  sufficient.  The  temperature  may  be  42.70.4 
—  105°  C.  from  normal,  that  is,  the  cold  resistance  would  be 
near  enough.  We  might  even  use  the  stated  resistance  with- 
out measurement.  Hence  also  the  error  from  this  source  can 
be  easily  rendered  of  negligible  amount,  and  we  may  use  n  =  5 
instead  of  n  =  7  as  above,  so  that  Ae/  \fn  would  become 
0.0022/2.2  =  o.ooio  if  the  discussion  were  to  be  revised. 
The  values  admissible  for  6v,  6c,  etc.,  would  then  be  corre- 
spondingly increased.  The  three  sections  of  this  paragraph 
answer  questions  (b)  and  (c)  of  the  problem. 

(d)  The  results  above  obtained  could  be  applied  without 
material  error  to  any  motor  not  differing  from  this  one  by 
more  than  about  io  per  cent  in  the  rated  values  of  F,  C,  Ra  , 
and  5,  or  by  more  than  40  per  cent  in  Rf. 

Note  that  if  the  efficiency  for  other  than  full  load  is  to  be 
found  with  the  same  or  any  given  precision,  a  solution  similar 
to  the  above  must  be  made  by  substituting  the  corresponding 
values  of  C  and  Ra  for  the  desired  load,  e.g.,  20  amp.  (or  more 
exactly  22  amp.)  and  about  0.14  ohms  for  half  load. 


130  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

Example  XXXIII.—  Cradle  Dynamometer.  —  The  princi- 
ple and  operation  of  the  Brackett  cradle  dynamometer  are  so 
well  known  that  a  brief  description  is  sufficient.  The  cradle 
consists  of  a  platform  suspended  at  each  end  from  a  horizontal 
knife-edge,  both  being  in  the  same  line.  The  dynamo  (gen- 
erator or  motor)  is  placed  securely  upon  the  platform  and 
adjusted  until  the  axis  of  rotation  of  its  shaft  is  coincident 
with  the  line  of  the  knife-edges.  The  centre  of  gravity  of  the 
whole  system  is  then  raised  or  lowered  by  means  of  auxiliary 
weights  until  just  below  the  axis.  The  system  then  oscillates 
slowly  and  sensitively  like  a  balance.  When  the  dynamo  is  in 
operation  the  mechanical  power  applied  to  or  given  out  by  its 
pulley  tends,  through  the  magnetic  reaction  between  the  arma- 
ture and  fields,  to  rotate  the  machine  and  therefore  the  whole 
system.  This  tendency  is  counterbalanced  by  a  weight  applied 
to  a  horizontal  lever  arm  projecting  from  the  dynamometer 
at  right  angles  to  the  axis.  The  weight  or  its  distance  or  both 
are  adjusted  until  a  balance  is  obtained  when  this  weight  w 
into  its  horizontal  distance  /  from  the  axis  gives  the  rotary 
moment  of  the  system,  and  therefore  that  applied  to  or  given 
out  by  the  dynamo.  From  this  and  the  speed  of  the  dynamo 
the  power  can  be  computed. 

Problem.  —  The  commercial  efficiency  at  full  load  of  a  certain 
generator  is  to  be  measured  by  a  cradle  dynamometer.  The 
dynamo  is  rated  at  75  volts,  60  amperes,  and  1400  rev.  per 
min.,  and  it  has  probably  an  efficiency  of  about  90  per  cent. 
The  diameter  of  its  pulley  is  2R  =  10  inches.  The  dynamo 
and  dynamometer  together  weigh  about  3000  Ibs.  The  length 
of  the  arm  to  carry  the  dynamometer  weights  is  /  =  3.4  ft. 
Required  in  advance  of  the  test  a  precision  discussion  of  the 
proposed  measurement  and  of  the  dynamometer.  The  value 
of  E  is  desired  to  one  per  cent. 

Solution.  —  First.  Precision  necessary  in  each  measured  com- 
ponent. The  expression  for  the  efficiency  in  terms  of  the 
measured  components  is 

O       CV   33000 


—>    —,  — 

746    27tlnw 


[109] 


CRADLE  DYNAMOMETER.  131 

The  measured  quantities  are  C,  V,  /,  w,  and  w  ;  .*.  n  =  5.     By 
[53]  we  have  for  equal  effects 


=  7S    T-    =  7|  X  0.01  =  0.0045. 

Each  of  these  components  then  must  be  measured  to  about 
0.45  per  cent. 

By  the  rules  for  significant  figures  the  constant  it  must  be 
carried  to  5  places,  i.e.,  3.1416  must  be  used.  This  is,  however, 
in  excess  of  the  strict  requirement  in  this  case,  as  will  almost 
always  be  true  when  those  rules  are  applied,  since  they  are 
framed  to  cover  the  worst  possible  case.  Applying  the  cri- 
terion for  constants  p.  70,  we  have,  to  be  negligible, 


—  <  ~^r  TT  <  a001 5  ; 

Hence  3.14  would  in  this  case  be  close  enough.  In  the  com- 
putation of  E  evidently,  by  the  rules,  5  places  must  be  re- 
tained in  each  component  factor,  so  that  retaining  n  =  3.1416 
does  not  materially  increase  the  labor  of  computation.  The 
constant  746  (see  next  page)  varies  with  the  force  of  gravitation, 
and  therefore  must  be  corrected  for  latitude  and  elevation.  By 
the  criterion  for  constants  it  must  be  exact  to 

(746)  =a°°15- 

The  change  in  g,  and  hence  in  the  constant,  is  only  about  0.003 
between  Edinburgh  and  the  equator,  and  is  only  at  the  rate  of 
about  o.oi  per  cent  per  1000  ft.  of  elevation,  so  that  these  cor- 
rections are  negligible  in  the  present  case.  There  is,  however, 
a  constant  error  of  about  0.3  per  cent  in  the  legal  ohm  and 
volt  of  1884,  so  that  the  constant  746  which  is  calculated  for 
the  theoretical  volt  is  about  0.3  per  cent  too  large  if  legal 
units  are  employed.  For  the  present  case  this  is  barely  worth 


132  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

considering.  It  might,  however,  be  worth  while  to  employ 
746  (i  —  0.003)  =  744-  instead  of  746.* 

Second.  Errors  of  Method,  i.e.,  Errors  and  adjustments  of 
the  cradle  dynamometer.  It  will  be  seen  as  we  proceed  that 
there  are  four  of  these  (.'.  p  =  4),  and  we  wish  to  determine 
how  small  each  error  or  how  close  each  adjustment  must  be  in 
order  that  the  error  shall  be  of  negligible  effect.  By  [80]  the 

admissible  limit  will  be  such  that  the  resulting  error  — —-  in  E 

I      AE 
shall   be  <  --  ~p—7^  <  0.0017.     As  the  effect  of  these  enters 

through    the    factor   /  =  -          -  which   contains    3    of   the    5 

33  ooo 

measured  components,  our  discussion  will  incidentally  give  us 
the  closeness  of  adjustment,  etc.,  in  the  dynamometer  necessary 
for  measuring  the  mechanical  power  with  it  to  ^3/5  X  o.oi  — 
0.77%,  or  about  f  of  one  per  cent,  with  the  given  load  and  power. 
The  adjustments  and  errors  referred  to  are  as  follows  : 
First,  two,  (a)  and  (b),  which  enter  however  rigid  may  be  the 
construction  of  the  dynamometer.  Second,  two,  (c)  and  (d), 
which  arise  from  yielding  of  the  structure,  i.e.,  from  want  of 
perfect  rigidity  of  construction  and  of  attachment  of  parts. 

*This  constant  746  for  reducing  watts  to  h.p.  is  derived  as  follows: 
i  Ib.  =  13  825^  dynes. 

g  =  980.6  —  2. 5  cos  2A  —  o.oo  ooo  3>4, 

where  units  are  c.  g.  s.,  /I  =  latitude,  h  =  ht.  above  sea  in  cm.  Thus  for  lat. 
45°  at  sea-level  £•  =  980.6  and 

i  h.p.  =  550  X  13  825  X  980.6  =  7.456  X  io9  ergs  per  sec. 
Now  i  volt-ampere  or  i  watt  by  definition  =  io8  X  io-1  =  io1  ergs  per  sec. 
.'.  i  h.p.  =  745.6  watts  at  45°,  sea-level, 

"     =  745.5      "     "  Boston,  sea-level, 
"     =745-9      "     "  place  where  g  —  981. 

The  latter  value  or  746  which  is  sensibly  equal  to  it  is  ordinarily 'adopted.  But 
it  is  important  to  remark  as  above  shown  that  these  values  are  about  0.3  per 
cent  too  large  if  the  legal  units  of  1884  are  used. 


CRADLE  DYNAMOMETER.  133 

(a)  Zero  or  index  error.  During  the  run  it  is  impossible,  of 
course,  to  maintain  the  dynamometer  at  its  normal  position  of 
equilibrium,  owing  to  continual  slight  fluctuations  in  power,  to 
jarring,  and  to  the  swinging  of  the  machine.  This  position  is 
usually  indicated  by  the  position  of  a  pointer  or  index  carried 
by  the  lever-arm  of  the  dynamometer.  Thev  reading  of  this 
index  upon  a  scale  is  taken  with  the  machine  at  rest,  and 
during  the  run  the  index  should  be  maintained  at  this  point  of 
rest.  As  this  cannot  be  done  exactly  let  n  (in  divisions  of  the 
scale)  denote  the  average  value  of  the  index  error.  Then  the 
dynamometer  with  dynamo  in  place  must  have  sufficient 
sensitiveness  so  that  the  fractional  error  in  the  weight  w  at  / 
due  to  this  index  error  shall  be  negligible. 

To  find  this  limit  let  a  weight  p  be  found  by  trial  which  if 
put  on  at  /  will  deflect  the  index  by  one  division.  Then  the 
sensitiveness  of  the  system  is  //,  i.e.,  this  is,  the  rotary  mo- 
ment corresponding  to  one  division.  The  fractional  index 
error  in  the  total  measured  moment  wl  found  during  the  run 
will  then  be  npl/wl.  To  be  negligible  this  must  be  ' 

npl  __  w 

^<  0.0017;     .'./  -0.0017  -. 

Suppose  that  inspection  of  the  apparatus  shows  that  n  will  be 
about  0.25  division.  We  require  also  the  value  of  w,  which 
may  be  found  as  follows.  The  output  of  the  dynamo  is  75  x 
60  ==  4500.  watts,  which  is  4500/746  =  6.0  h.p.  The  efficiency 
of  the  generator  being  about  90  per  cent,  the  mechanical 
power  will  be  6.0/0.9  —  6-7  h.p. 

2nlnw  6.7  X  33  ooo. 

'  =  =  2X3.IX34XI400.  =  7-5 


Thus  the  sensitiveness  in  order  that  the  index  error  may  be 
negligible  must  be 

/  ^0.0017-^-=  0.050  Ib.  =  0.8  oz. 
This  sensitiveness  may  usually  be  easily  reached  or  exceeded. 


134  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

(b)  Shaft  to  line  of  knife-edges.  The  axis  of  the  shaft  must 
lie  strictly  in  the  line  joining  the  knife-edges  supporting  the 
cradle.  Otherwise  the  resultant  pull  of  the  belt  on  the  dynamo 
pulley,  since  it  acts  through  and  at  right  angles  to  this  axis, 
will  tend  to  rotate  the  dynamometer.  Thus  if  the  axis  is  dis- 
placed parallel  to  the  knife-edge  line  by  a  distance  d,  and  if  2a 
is  the  sum  of  the  tensions  in  the  two  sides  of  the  belt  assumed 
parallel,  then  the  resultant  belt  pull  is  2a  and  the  moment  of 
rotation  which  this  exerts  upon  the  dynamometer  is 

2ad  sin  0,    .     .     .     .     v    .     .     [110} 

where  6  is  the  angle  between  the  directions  of  d  and  2a.  If 
the  index  is  adjusted  to  zero,  or  its  position  of  rest  taken,  with 
the  belt  on  and  tight,  this  erroneous  moment  would  be  counter- 
balanced once  for  all  by  the  adjusting  weights  of  the  dynamom- 
eter, provided  that  2<z,  i.e.,  the  belt  tension,  did  not  change 
during  the  run.  This  constancy  is,  however,  hopeless.  The 
tension  will  change  sensibly  between  running  and  rest,  and 
more  or  less  progressive  change  of  length  and  consequently  of 
tension  will  occur  during  the  run.  Let  us  then  first  see  how 
small  d  must  be  in  order  that  a  change  equal  to  the  whole 
belt  pull  2a  (i.e.,  one  due  to  throwing  on  and  off  the  belt)  shall 
be  of  negligible  effect.  In  the  present  case  with  a  leather  belt 
on  an  iron  pulley  we  may  assume,  as  shown  below,*  2a  =  360. 
Ibs.  Further,  as  B  may  have  any  value,  let  us  solve  for  the 

2 

average  value  -  =  0.65  which  sin  0  would  have  in  the  long  run 

*  The  value  of  2a  may  be  calculated  as  follows  :  Let  /  =  tension  in  tight 
side  of  running  belt,  and  s  =  that  in  slack  side,  and  a  =  one  half  the  sum  of 
these.  Then 

/^g  +  K/-,) 

s       a-\(t-  s) 

For  a  leather  belt  on  an  iron  pulley  t/s  cannot  be  more  than  about  f  without 
undue  slip,  and  on  tight  dynamo  belts  it  is  likely  to  be  much  less  than  this. 
Equating  to  f  and  solving  gives  a  =  2(t  —  s).  A  value  of  a  =  3(/  —  s)  would  be 
of  more  frequent  occurrence. 

In  this  problem  we  have  obviously 


CRADLE  DYNAMOMETER.  135 

if  all  values  of  6  were  equally  probable.     The  fractional  error 
in  /  and  therefore  in  E  due  to  this  cause  is  then 

2ad  sin  0/wl,      ......     [HI] 

which  to  be  negligible  must  be  =  0.0017.     Whence  to  be  of 
negligible  effect  d  must  be 

wl  7.5  X  3-4 

d  <  ao°I7^i^  =  a001  73-6^0:55  =  ao°019  ft 

=  0.0023  inch. 

It  is  obvious  that  no  such  adjustment  as  this  can  be  made,  and 
therefore  the  index  must  be  adjusted  to  zero  or  the  position 
of  rest  taken  with  the  belt  on  and  tight.     If  this  is  done,  then 
the   error  from  the   steady  pull  will   be  eliminated,  and  all 
rapidly  oscillating  changes  will  obviously  merely  cause  oscilla- 
tions of  the  dynamometer  and  will  eliminate  themselves.     But 
progressive  changes  will  cause  error.    How  much  of  such  change 
will  occur  ?     The  answer  must  be  largely  a  matter  of  estimate. 
Let  us  assume  that  a  change  of  one  tenth  of  2a,  i.e.,  of  36  Ibs., 
is  as  large  as  we  need  expect.     Then  for  this  to  be  of  negligible 
effect,  we  must  have 


3     =  0.0019  ft.  =  0.023  inch. 


As  close  an  adjustment  as  this  is  hardly  to  be  expected  on 
such  an  apparatus.  Hence  this  source  of  error  cannot  probably 
be  rendered  negligible,  and  its  effect  on  E  may  even  exceed 
some  of  the  other  errors  of  measurement. 


-j-)^  =  w/;     .-.  t-s=  7>5  *3'4  =60.  Ibs. 
I* 

.*.  a  =  180.  Ibs.,     and     2a  =  360.  Ibs. 


136  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

(c)  The  pull  of  the  belt  will  also  cause  error  in  two  other 
ways.  As  neither  the  dynamometer  nor  dynamo  can  be  made 
perfectly  rigid  and  inflexible,  and  as  the  dynamo  cannot  be 
attached  to  the  dynamometer  with  perfect  firmness,  the  pull  of 
the  belt  will  strain  the  structure  somewhat.  If  the  shaft  of  the 
dynamo  be  aligned  to  the  knife-edges  with  the  belt  off,  then 
when  the  belt  is  thrown  on,  this  adjustment  will  be  disturbed 
owing  to  the  yielding  of  the  structure.  Both  the  axis  of  rota- 
tion of  the  shaft  and  the  centre  of  gravity  of  the  whole  system 
will  be  thrown  out  of  adjustment  thereby.  There  will  thus  be 
introduced  two  erroneous  moments,  one  of  the  same  character 
as  (b),  that  is  due  to  the  resultant  belt-pull  acting  through  a 
point  outside  of  the  line  of  the  knife-edges ;  the  other  due  to 
the  weight  of  the  system  acting  through  the  displaced  centre 
of  gravity.  The  first  of  these  will  be  now  discussed,  the  second 
under  the  heading  (d). 

How  small  must  the  displacement  d  of  the  shaft  by  the 
belt-pull  be,  in  order  that  the  error  due  to  the  pull  shall  be 
negligible?  Obviously  the  solution  will  be  precisely  the  same 
as  in  (£),  except  with  respect  to  the  value  of  0.  The  displace- 
ment of  the  axis  in  this  case  will  not  be  equally  likely  to  occur 
in  any  direction,  neither  will  it  be  always  or  in  general  in  the 
direction  of  the  belt-pull.  The  direction  will  be  determined  by 
the  constraint  of  the  various  parts  of  the  structure,  but  will 
tend  to  be  most  largely  in  the  direction  of  the  pull.  No 
general  average  value  of  sin  6  can  then  be  stated,  but  the  aver- 

2 

age  would  be  less  than  the  —  employed  in  (b).     We  shall  there- 
fore obtain  an  excessive   but  safe  limit  for  d  by  using  -  as 

7f 

before.     The  erroneous  moment  if  the  shaft  is  displaced  by  d 
is  then 

2ad  sin  9, 

and  to  be  negligible  the  value  of  d  must  then  as  before  be 
d  =  0.0023  inch. 


CRADLE  DYNAMOMETER.  137 

This  amount  of  displacement  must  then  not  occur  with  the 
total  belt-pull  if  the  zero  adjustment  is  made  only  with  the 
belt  off,  or  must  not  occur  with  one  tenth  of  2/2  if  the  zero  ad- 
justment is  made  with  the  belt  on,  and  we  again  accept  that  as 
a  limiting  value  of  the  progressive  change  in  belt-pull.  It  is 
doubtful  whether  the  rigidity  called  for  by  this  limit  can  be 
reached,  and  whether  this  error  will  not  enter  with  at  least  the 
same  magnitude  as  that  under  (b). 

(d]  Let  h  denote  the  horizontal  component  of  the  displace- 
ment of  the  centre  of  gravity.  Then  the  weight  W  of  the 
whole  system  acting  at  right  angles  to  this  will  cause  an 
erroneous  moment  tending  to  rotate  the  dynamometer  whose 
amount  will  be  Wh,  which  must  be  counterpoised  on  the  lever 
arm.  The  fractional  error  in  the  power  measurement  will 
therefore  be 

Wh 

wf 

which  to  be  negligible  must  be  =  0.0017.  Hence  for  negligi- 
bility 

h  <o.ooi7Tr>  =  0.0017- '-  =0.000015  ft.=o.oo  018  in. 

This  amount  of  horizontal  displacement  must  then  not  be  pro- 
duced by  whatever  progressive  change  of  belt-pull  may  occur 
(e.g.,  one  tenth  of  2a  =  36  Ibs.  as  above),  or  by  the  total  belt- 
pull  if  the  zero  point  is  not  taken  with  the  belt  on.  Such 
rigidity  is  not  to  be  hoped  for  with  a  horizontal  belt-pull  or 
with  the  belt  in  any  direction  but  the  vertical.  The  belt  must 
therefore  be  vertical  and  should  preferably  run  downward,  as 
this  tends  to  least  distortion  of  the  structure.  Just  how  much 
the  centre  of  gravity  would  be  displaced  by  any  given  pull 
with  any  given  apparatus  is  hardly  determinable.  The  only 
thing  to  be  done  is  to  have  the  dynamometer  designed  for 
great  rigidity  and  lightness,  and  to  see  that  the  dynamo 
and  all  attachments  are  well  secured.  An  important  point  in 
the  design  is  the  stiffness  of  the  upright  screws  which  carry  the 
counterpoise  blocks  above  the  axis  of  rotation. 


138  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

The  uncertainty  introduced  by  this  source  of  error  (d)  is 
probably  the  greatest  entering  into  the  use  of  the  dynamometer. 

Summary. — From  the  foregoing  considerations  then  we  see 
that  :— 

Dynamo  shaft  must  be  adjusted  to  line  of  knife-edges  with 
utmost  care. 

Zero  reading  or  adjustment  must  be  made  with  belt  on  and 
tight. 

Dynamometer  must  be  of  extremely  rigid  design,  and  as 
light  as  possible ;  and  all  attachments  must  be  firm. 

Belt  must  run  vertically  downward  from  the  dynamo. 

Belt-pull  must  be  as  slight  as  possible,  and  therefore  belt 
must  be  slack  and  pulley  of  large  diameter.  It  would  be  better 
to  run  the  dynamo  by  a  couple  without  thrust. 

When  all  these  are  attended  to  it  is  doubtful  whether  a 
measurement  of  power  accurate  to  i  per  cent  can  be  obtained. 

Example  XXXIV.  —  Tangent  Galvanometer.  —  Problem. 
Given  a  good  primary  tangent  galvanometer  of  the  type 
described  below;  required  a  preliminary  discussion  to  show 
what  precision  can  be  obtained  in  its  use. 

Description. — Coil  of  n  turns  having  a  mean  radius  r  of 
about  20  cm.,  and  being  of  rectangular  section  of  breadth 
2b  =  2.0  cm.,  and  depth  2d  =  2.4  cm.,  about.  Needle,  a  bundle 
of  bits  of  watch-spring  the  distance  between  poles  being 
2/  =  I  cm.,  approximately,  suspended  by  a  single  silk  fibre,  the 
coefficient  of  torsion  being  6.  Index  attached  to  the  needle 
and  consisting  of  a  bit  of  spun  black  glass.  This  moves  over 
a  graduated  circle  of  about  10  cm.  diameter,  divided  into 
degrees  and  secured  to  a  circular  mirror  to  reduce  parallax. 
Readings  taken  to  o°.i  by  eye  estimation,  both  ends  of  index 
and  reversals  of  current  being  read,  making  four  readings  to 
be  averaged.  The  expression  for  the  current  corresponding 
to  an  observed  mean  deflection  0°  is 


TANGENT  GALVANOMETER.  139 

where  H  is  the  horizontal  component  of  the  intensity  of  the 
earth's  magnetic  field  at  the  needle,  G  is  the  constant  of  the 
galvanometer,  and  the  several  ( )  represent  correction  factors 
for  coil  section,  length  of  needle,  torsion,  etc. 

Solution. — The  statement  of  the  problem  assigns  no  definite 
value  for  the  precision  to  be  attained,  but  the  requirement  is 
to  ascertain  what  precision  may  be  obtained.  As  convenient 
a  way  as  any  of  attacking  this  problem  is  to  solve  for  the  value 
of  dH/H,  which  will  produce  separately  an  error  dC/C,  of  some 
suitable  amount,  e.g.,  o.ooi,  and  to  make  a  similar  solution  for 
the  same  value  of  6C/C  for  each  measured  component  and  for 
the  various  corrections,  adjustments,  etc.  These  results  can 
then  be  conveniently  discussed  to  ascertain  whether  they  can 
be  attained  or  exceeded,  and  a  final  summing  up  then  made  to 
see  what  resultant  precision  AC/C  is  attainable. 

We  have  first  to  prepare  the  expression  for  C  for  discussion. 
As  G  is  calculated  from  the  measured  dimensions  of  the  coil, 
it  must  be  expressed  in  terms  of  them.  Suppose  the  radius  of 
the  coil  to  be  found  by  measuring  its  inner  and  outer  circum- 
ference at  the  time  of  winding.  It  will  be  exact  enough  to 
regard  this  as  one  measurement  of  its  circumference  s.  Then 


c 


27t 


The  quantity  H  is  the  result  of  a  complex  indirect  meas- 
urement. As  we  do  not  care  to  complicate  the  discussion  of 
the  present  problem  by  introducing  the  detailed  discussion  of 
H,  we  will  treat  it  as  a  directly  measured  component  and  leave 
its  further  consideration  for  separate  treatment.  The  expres- 
sion then  in  proper  form  is 


140  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

The  several  parentheses  will  be  written  in  full  below.  The 
simplest  method  of  treatment  is  by  separation  into  factors 
which  are  functions  of  one  or  more  components,  as  in  [75]. 
These  will  be  denoted  by  flt  ft,  etc.  They  are  /,  =  H9 

/,  =  *,/.  =  »~',/4  =  tan  0»/.  =  (  )>  etc. 

/,  =  H.  —  By  the  conditions  of  the  problem  then 


—  =  -=0.001. 

The  measurement  of  H  with  this  accuracy  is  difficult  and 
laborious  ;  moreover,  the  diurnal  and  local  fluctuations  are  of 
about  this  order  of  magnitude,  though  the  latter  may  be  much 
larger.  It  is  hardly  practicable  in  any  ordinary  laboratory 
work  to  depend  upon  H  as  constant  within  about  twice  this 
limit,  or  0.2  per  cent,  and  this  only  under  favorable  conditions. 
By  a  good  magnetometer  it  may  be  measured  to  less  than  0.2 
per  cent.  In  merely  relative  measurements  of  C  the  absolute 
value  of  H  need  not  be  known  and  the  effect  only  of  variations 
in  H  enters. 

R  S>  f 

/,  =  s.  —  We  must  have  —  =  -~-  =  0.00  1.     As  r  =  20  cm., 

S  Ls 

s  =  2nr  —  1  20.  cm.  /.  ds  =  0.0015  =  0.12  cm.  The  error  in 
r  involves  not  only  errors  in  the  measurement  of  s,  but  irregu- 
larity in  the  distribution  of  the  convolutions  of  the  coil.  The 
measurement  of  s  can  doubtless  be  made  closer  than  0.12  cm., 
but  the  uncertainty  with  respect  to  irregular  winding  owing  to 
varying  tension,  to  varying  thickness  of  insulation,  etc.,  will 
probably  not  be  much  less  than  that  amount.  We  may  prob- 
ably count  on  this  limit  as  about  what  is  practically  attainable 
in  a  good  coil  carefully  wound  into  a  channel. 

If  the  coil  is  not  a  true  circle  but  is  more  or  less  elliptical, 
the  expression  above  given  for  G  will  not  be  exact.  To  find 
the  amount  of  error  for  a  given  ellipticity  it  would  be  necessary 
to  deduce  an  expression  for  the  field  at  the  centre  of  an  ellip- 
tical circuit  which  cannot  readily  be  done.  It  is  easy  to  see, 
however,  that  the  field  does  not  change  materially  for  slight 


TANGENT  GALVANOMETER.  141 

eccentricity,  for  if  a  circular  circuit  be  gradually  deformed  into 
an  ellipse  the  flattened  sides  approach  the  centre  at  first  at 
sensibly  the  same  rate  as  that  at  which  the  bulging  ends  recede. 

/3  =  n.  —  This  is  necessarily  a  whole  number  and  not  subject 
to  error  except  through  mistake  in  counting.  A  mistake  of 
one  turn  in  a  thousand  would  correspond  to  the  assigned 
limit  of  3C/C,  but  as  the  value  of  n  will  seldom  be  as  large  as 
1000,  no  mistake  is  allowable. 

/4  r=  tan  0.  —  We  will  make  the  solution  for  0  =  45°,  but 
the  result  will  apply  without  sensible  error  to  any  deflection 
between  30°  and  60°  as  shown  later.- 

d  tan  0       dC 


.*.  d  tan  0  =  o.ooi 

From  this  we  have  to  determine  the  corresponding  value 
of  £0  by  [34]. 


£0  =  d  tan  0/ 


d  tan  0 
d<t> 


0  2      ,  I 

—  =  sec   0  = 


cos30' 

/.  <J0  =  o.ooi  cosa  0  =  o.ooi  X  0.50 

=  o.oo  050  in  radian  measure. 

.-.  £0  =  o°.03. 

The  value  of  0  is  a  mean  of  four  readings  in  which  the 
tenths  of  a  degree  are  estimated  by  the  eye.  These  estima- 
tions if  properly  made  will  always  give  the  nearest  tenth.  The 
extreme  error  of  estimation  will  therefore  be  -f-  o°.O5  and 
—  O°.O5,  and  the  error  will  be  equally  likely  to  have  any  value 
between  these  limits.  The  average  error  of  a  single  estima- 
tion will  be  (page  21)  o°.O25,  and  of  the  mean  of  four  will  be 
O.O25/  V^.  =  O°.OI3,  which  is  negligible  compared  with  the  value 


142  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

O°.O3  above  deduced.  But  there  are  other  sources  of  error  in 
0,  viz.,  irregularities  in  graduation,  eccentricity,  parallax,  tor- 
sion, coils  out  of  vertical  plane,  coils  out  of  magnetic  meridian. 
Of  these  the  first  is  partly  eliminated  by  the  reading  at  four 
different  points  on  the  circle,  the  second  by  reading  both  ends 
of  the  index,  the  third  by  bringing  the  eye  when  reading  into 
a  position  where  the  index  just  covers  its  reflection  in  the 
mirror;  the  fourth,  fifth,  and  sixth  are  corrected  for  by  the  cor- 
responding correction  terms  which  are  separately  discussed 
below.  The  residual  errors  from  these  four  corrections  are 
classed  separately  and  therefore  do  not  need  to  be  regarded 
as  augmenting  #0.  The  errors  from  the  first  three  sources 
may  obviously  be  of  any  amount  according  to  construction  of 
instrument,  but  need  not  exceed  the  limit  of  o°.O3  which  may 
be  regarded  as  practically  attainable. 

/5  =  (  I  -|  ---  a  ---  j-J-  —  This  is    the    correction  term   for 

V         2  r       3  r  ' 

finite  dimensions  of  the  rectangular  coil  section,  and  is  suffi- 
ciently close  where  the  depth  does  not  exceed  one  tenth  of 
the  radius.  It  is  in  reality  a  correction  to  the  value  of  G. 
2b  =  breadth,  2d  =  depth.  The  term  involves  r  and  therefore 
the  measured  components,  but  as  the  (  )  differs  from  unity  by 
only  one  or  two  per  cent  at  most  it  may  be  omitted  in  dis- 
cussing 8sy  as  was  done  above,  and  r  may  now  be  treated  as  a 
constant.  This  factor  contains  then  two  measured  quantities 
2b  and  2d.  The  value  of  df  \/J  \  corresponding  to  equal  effects 
for  each  must  then  be  o.ooio  V~2  =  0.0014,  corresponding  to 
which  we  have  to  find  the  values  of  6(2b)  and  d(2d).  As/6  is 
sensibly  =  i  we  have 


=  0.0014. 

By  [37].  [45],  and  [34] 


=  0.0020  /  -7  =  0.0020  /—  r  =  °'8°  cm» 


TANGENT  GALVANOMETER.  143 

It  is  obvious  that  this  limit  is  needlessly  large.  A  negligible 
amount  would  be  one  third  of  this,  viz.,  0.27  cm.,  and  we  can 
easily  measure  the  depth  much  closer  than  this.  If  then  the 
breadth  measurement  be  made  to  3  mm.,  its  residual  error  will 
be  negligible. 

Similarly  for  the  depth  measurement 


dd 
2d 


I2 

=  o.oo2cy  -^r  =  i. 


o  cm. 


Therefore  if  the  depth  be  measured  to  one  third  of  this,  i.e., 
to  3  mm.,  the  residual  error  will  be  entirely  negligible. 

The  correction  itself  would  be  negligible  when  the  coil 
•section  was  such  that  2b  <  0.4  cm.  and  2d  <  0.6  cm. 

Inspection  of  the  form  of  the  correction  shows  that  if  the 
coil  section  be  so  designed  that 


. 
2  r*  ~  3  ra 

the  correction  will  vanish.     Solving  gives 

tf       2  2b      5 

• 


If  the  coil  be  wound  to  these  relative  dimensions  then  the 
correction  may  be  omitted.  Obviously  the  dimensions  must 
be  adhered  to  within  the  limits  d(2&)  =  4  mm.  and  $(2d)  —  6 
mm.  in  this  case,  or  within  limits  having  the  proportion  of 
d(2b)  :  6(2d)  =  2:3  in  any  case.  These  limits  must  be  con- 
sidered not  merely  as  referring  to  the  outside  dimensions  of 
the  coils,  but  to  the  density  of  winding  as  well.  If  the  wind- 
ing is  not  in  "  square  order,"  but  is  more  dense  in  the  depth 
than  in  the  breadth  of  the  coil,  the  correction  by  the  above 
formula  with  respect  to  the  breadth  will  be  too  great  relatively 


144  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

to  the  depth  about  in  the  proportion  of  the  relative  densities, 
and  a  corresponding  allowance  must  be  made. 

/      3  r      15  r         \ 

/.  =  f I  — s- -| r  sin2  01. — The     expression    for    G 

\         4  T         4    T  ' 

gives  the  field  at  the  centre  of  the  coil  due  to  a  unit  (c.  g.  s.) 
of  current  in  the  coil.  The  needle  has,  however,  a  finite  length 
2l,  and  its  poles  therefore  lie  in  a  field  which  when  0  is  nearly 
zero  is  slightly  less  intense  than  at  the  centre,  and  which 
increases  with  0.  The  second  term  in  the  correction  takes 
account  of  the  fact  of  the  less  intensity,  the  third  term  of  the 
want  of  uniformity  of  the  field.  The  centre  of  the  needle  is, 
of  course,  assumed  to  be  at  the  centre  of  the  coil.  The  ex- 
pression is  sufficiently  exact  when  2l  is  less  than  one  tenth  of 
the  radius  of  the  coil. 

Since  the  value  of  /.  in  an  extreme  case  differs  from  unity 
by  only  one  or  two  per  cent,  it  may  be  omitted  when  r  and  0 
are  being  discussed,  as  was  done  above.  We  have  then  only 
one  component  here,  viz.,  2l.  Then 


As  fi  =  I  very  nearly 

4 

A  f 

<5>/6  —  -7-'  =  o.ooio  approx. 


Inspection  shows  that  the  value  of  /6  is  greatest  for  0  ==  60° 
if  the  galvanometer  is  used  (for  reasons  later  stated)  only 
between  30°  and  60°.  We  will  therefore  solve  for  the  worst 
case.  Substituting  gives 


d(2l)  =  0.0020/-J-  =  0.40 


200 


TANGENT  GALVANOMETER.  145 

Greater  accuracy  than  this  is  easily  attainable.  0.40/3 
—  0.13  cm.  would  be  negligible,  and  this  can  be  reached.  In 
order  that  the  correction  may  be  wholly  negligible  /  must 

be  such  that    —  -  ^-  +  —  —r  sin8  0  =  o.      For  0  =  60°,  this 
4  *         4  r 

gives  2/  =  — ,  or,  as  r  —  20,  2/=  0,5  cm.  =  5  mm.     But   the 

4° 

needle  cannot  be  made  so  short  as  this,  on  account  of  torsion, 
so  that  the  correction  itself  at  60°  can  never  be  negligible.  But 
the  length  can  be  measured  accurately  enough  so  that  the 
residual  error  shall  be  negligible.  It  should  be  noted  that  2/ 
the  pole  distance  is  about  0.85  of  the  total  length  of  the  needle 

if  this  is  a  thin  rectangular  prism. 

3/8        jc/2 

The  correction  obviously  vanishes  when  -  -j  =  — =  —  sina  0, 

4r         4  r 

which  solved  for  0  gives  0  =  26°.6.  The  correction  is  nega- 
tive below  and  positive  above  that  angle.  As  far  as  concerns 
this  error  alone,  26.°  would  then  be  a  favorable  angle  at  which 
to  use  the  instrument,  but  there  are  other  considerations  which 
outweigh  this,  as  will  be  presently  shown.  At  45°  the  correc- 
tion term 'for  the  case  in  hand  would  be  1.0014  which  is  four 
times  the  negligible  amount. 

/7  =  (i  +  ^  *  ~^~y  ~2Z\— The  expression  for /.  is  based 

\  2  Y  i 

on  the  assumption  that  the  centre  of  the  needle  is  at  the  coil 
centre.  This  adjustment,  especially  with  a  suspended  needle, 
cannot  be  made  with  exactness,  and  it  is  necessary  to  know 
how  closely  it  must  be  made.  The  above  expression  gives  the 
correction  factor  to  be  applied  when  the  needle  is  slightly  out 
of  centre,  x  is  the  horizontal  component  and  y  the  vertical 
component  of  the  displacement  of  the  needle  in  the  plane  of 
the  coil,  and  z  is  the  displacement  along  the  axis  of  the  coil. 
In  other  words,  relatively  to  the  centre  of  the  coil  the  coordi- 
nates of  the  centre  of  the  magnet  in  its  displaced  position  are: 
z  along  the  axis  of  the  coil,  and  x  and  y  at  right  angles  to  this 
and  to  each  other.  As  a  basis  for  numerical  solution  let  us 
assume  that  we  can  always  set  the  needle  so  near  the  centre 
that  neither  x,  y,  nor  z  shall  exceed  a  given  distance  a  more  than 


146  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

once  in  a  thousand  times.  The  correction  will  be  a  maximum 
when  x  =  y  =  a  and  z  =  o,  or  when  x  =.y  =  o  and  z  =  a.  Let 
us  assume  further  that  the  law  of  accidental  placing  of  the 
needle  is  such  that  the  magnitude  of  the  correction  terms  will 
follow  the  general  law  of  distribution  of  deviations.  The  max- 
imum value  will  then  be  3— -  and  the  average  value  -  --.  This 

r*  ^r* 

assumption  is  surely  not  exact,  but  is  probably  sufficiently  cor- 
rect for  the  purpose.     What  value  then  must  a  have  to  render 
this  correction  negligible ;  for  obviously  we  cannot  well  measure 
;r,  y,  and  z  and  correct  for  it  each  time  we  use  the  instrument. 
We  have  then 


and  in  order  that  this  shall  be  negligible  we  must  have 

J*  =  1X000..  -  > 

For/,  must  not  exceed  i  ±  $  X  o.ooi. 

' .-.    a*  =  $  x  o.ooi  X  400  =  0.18  cm. ; 
a  =  0.42  cm. 

Hence  the  centering  will  be  close  enough  if  x,  y,  and  z  never 
exceed  0.42  cm.     This  is  easily  possible,  but  requires  care,  and 
should  always  be  attended  to  in  setting  up  the  instrument. 
For  the  same  reason  as  in  f6  and  ft1  r  is  here  treated  as  a 

constant. 

(/  n  \ 
I  -| — — — ). — This  is  the  correction  factor  for  torsion 
sin  0/ 

when  this  is  not  unduly  great ;  0  is  to  be  expressed  in  degrees. 
It  does  not,  however,  take  into  account  the  effect  of  initial 
torsion,  which  is  eliminated  by  the  process  of  reversal  of  the 
current.  Here  0  is  the  "coefficient  of  torsion."  If,  as  is 


TANGENT  GALVANOMETER.  147 

usual,  6  is  determined  by  reading  the  change  a  in  the  zero 
reading  produced  by  twisting  the  fibre  top  (or  bottom)  through 
360°,  we  have 

a          sin  a  ...       sin  a 

v=—  -  ,    or  sensibly      —  —  , 
360  —  a  360 

and  this  should  be  substituted  in  the  above  expression  to  pre- 
pare it  for  discussion,  since  a  is  the  measured  quantity.  Hence 


sin  0     360 


As  in  discussing/5  the  deflection  0  will  be  taken  at  60°,  since 
the  correction  then  has  its  largest  value. 
How  closely  must  a  be  measured  ? 


d(  )  0      cos  a  _         cos  a 

da  sin  0     360  5.2 

A  not  unusual  value  for  a  is  3°,  although  it  may  easily  be  made 
less.  For  a  =  3°, 

da  =  0.0010/0.19  =  0.0052  rad.,  or  O°.3O. 

This  would  correspond  to  6C/C  '=  o.ooi,  but  one  third  of  it, 
viz.,  o°.io  would  be  negligible,  and  as  this  is  easily  reached,  the 
torsion  can  easily  be  corrected  so  that  the  residual  error  shall 
be  negligible.  In  order  that  the  entire  correction  may  be 
omitted,  we  must  have  or  =  o°.  10,  which  can  be  attained,  but 
requires  an  exceptionally  fine  fibre  and  strong  magnetization 
of  the  needle.  Inasmuch  as  the  correction  for  length  of  needle 
cannot  be  rendered  negligible,  it  would  be  better  to  use  a 
longer  needle,  say  1.5  to  2.0  cm.,  thus  making  the  torsion 


148  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

negligible  and  throwing  all  the  correction  into  the  factor  for 
length  of  the  needle. 

/9  =  -  3.  —  The  plane  of  the  coils  should  be  vertical,  as 
cos  ft 

that  is  the  supposition  upon  which  the  law  of  tangents  is 
deduced.  If  the  coil  is  inclined  through  an  angle  ft,  we  know 
by  the  same  demonstration  as  for  the  cosine  galvanometer  that 


G     cos 


Let  us  inquire  what  value  of  ft  would  produce  a  negligible 
error,  as  we  merely  wish  for  a  guide  to  show  how  accurately  we 
must  level  the  instrument.  Perhaps  the  simplest  method  of 
solving  is  as  follows.  As  ft  is  small  we  may  write  cos  ft  =  I  —  x> 
where  x  is  a  small  fraction.  Hence 

=  i  +  x  approximately. 


I—  x 


To  be  negligible,  we  must  have 

x  =  0.00  033  ;     .-.  cos  ft  =  =  0.99  967 ;     .-.  ft  =  i°.5. 

This  is  easily  reached  in  levelling  by  plumb-line  or  otherwise, 
as  it  corresponds  to  a  displacement  of  the  top  of  the  coil 
beyond  the  bottom  by 

2r  sin  i°.5  =  1.0  cm., 

which  is  easily  perceptible. 

/io. — When  the  galvanometer  is  to  be  used,  the  plane  of  its 
coils  should  be  adjusted  into  the  magnetic  meridian.  This  is 
done  either  by  finding  the  meridian  by  means  of  an  auxiliary 
compass  and  setting  the  coil  to  correspond,  or  by  turning  the 
coil  about  a  vertical  axis  until  reversals  of  current  show  equal 


TANGENT  GALVANOMETER. 


149 


deflections  in  opposite  directions.  The  latter  method  is  inter- 
fered with  by  any  initial  torsion  which  may  be  present  in  the 
fibre.  Either  method  can,  of  course,  only  bring  the  coil  more 
or  less  closely  into  the  meridian,  and  it  is  essential  to  know 
how  closely  the  adjustment  ought  to  be  made. 

In  Fig.  i  let  NS  show  the  direction  of  the  magnetic  merid- 
ian. This  is  the  direction  in  which 
the  axis  of  the  undeflected  needle 
will  normally  stand.  Suppose  the 
centre  of  the  needle  to  be  at  P.  And 
let  QR  represent  the  direction  of  the 
plane  of  the  coils  making  an  angle 
OPQ  =  GO  with  the  meridian.  Let 
PO  represent  the  earth's  horizontal 
component  //,  and  PL  the  field  F 
produced  by  a  +  current  C  through 
the  coil.  Then  under  this  current  the 
resultant  field  will  be  PBy  and  the 
needle  will  set  in  that  direction,  its 
deflection  being  OPB  =  0,.  With 
the  same  current  reversed,  the  field 
will  be  equal  and  opposite  to  PF, 
and  the  resultant  field  will  be  PB ', 
and  the  deflection  02  —  OPB'.  In 
the  use  of  the  instrument  the  mean 
angle  is  employed,  viz.  £(0,  -f-  02).  The  deflection  02  on  the 
same  side  with  GO  is  obviously  greater  than  <pl  on  the  opposite 
side.  The  use  of  the  mean  angle  rests  on  the  assumption  that 
02  is  just  as  much  greater  as  0,  is  smaller  than  the  true  angle. 
This  assumption  is  in  general  not  exact,  but  is  more  nearly  true 
as  GO  is  smaller  and  as  0  is  more  nearly  45°,  being  true  for  that 
angle  whatever  the  value  of  GO,  as  will  be  shown  later.  It  re- 
mains then  to  ascertain  the  algebraic  relation  between  0t ,  02 , 
ft?,  and  the  true  angle  0  which  would  be  obtained  if  GO  were 
zero. 

In  Fig.  2  the  letters  correspond  with  Fig.   I.     Suppose  GO 
=  o.     Then  the  deflections  would  be  OP  A  and  OPA',  and  they 


FIG.  i. 


SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 


would  both  be  the  same,  and  each  equal  to  the  true  angle  0. 

Suppose  the  coils  to  be  inclined 
at  some  angle  GO  as  in  Fig.  i. 
Draw  through  O,  Fig.  2,  a  line 
IA  BOB'  making  any  angle  GO  with 
AOA',  and  lay  off  along  it  OB  = 
OA  =F  and  OB'  =  OA  =  OA' 
=  F.  Then  obviously  OB  and 
OB'  will  be  the  fields  due  to  the 
same  current  C  with  the  coils 
inclined  at  GO°  ;  PB  and/>£'  will 
FlG-  *  be  the  resultant  fields;  and  OPB 

will  be  0X  and  OPB'  will  be  0Q.      Prolong  PB'  to  meet  A  A' 

prolonged  in  C'.    PB  cuts  A  A'  at  C.     We  proceed  first  to 

find  expressions  for  tan  0,  and  tan  0a. 
In  the  triangle  OCB  we  have 


VC__OC__  sin  Ogf  _  sin  (90°  +  GO  +  0,)  _  cos  (a>  + 
""      sin  OCB  ~     ~  sin  (90°  +  0,)  cos  0, 

.        OC       F   OC      F  cos  (a?  +0,) 
-.  ---.—  .    -- 


tan  0  •  cos  ^  ~r_gV  _  tan  0.cos  &  _  tan  0-sin  co-tan  0, 
cos  0, 

/.  tan  0!  =  tan  0-cos  GO/(I  +  tan  0-sin  GO).   . 
In  the  triangle  OC'B'  we  have 


OC       OC  _  sin  Off' (7  _  sin  (90°  —  GO  +  0a)  _  cos  (GO  —  0a) 
O57  =  ~^~  :~  sin  OCB1  ~       sin  (90°  —  0a)  "  cos  0a 

OC_  __  F   OC  _  F_  cos  (6?  -  0a) 
H    ~  H'  F     ~  H       cos  0a 

=  tan  0- cos  G?  +  tan  0-sin  a?'tan  0a ; 


tan  0 


C°S 


COS   0a 

.'.  tan  0a  =  tan  0-cos  GO/(I  —  tan  0-sin  GO). 


TANGENT  GALVANOMETER.  151 

Hence 

tan  (0,  +  0.)  =    tan^  +  tan^ 
i  —  tan  01-tan01 

(tan  0-cos  G?  tan  0-cos  GJ  ]    /  [          tan2  0-cos2  c»  1 

I  -f-  tan  0-sinft?~'~  i—  tan0-sin  c»j  /  j        i—  tan2  0-sin2  a?J 

_  2  tan  0-cos  ft?        2  tan  J  (0,  -|-  0a) 
~i  -tan2  0       =  T^tan"  i(0t  +  02)  * 

/.[I  -tan9  1(0,  +  0J]  tan  0-cos  <y  =  (i-tan2  0)tan  i  (0,  +  0.), 
tan  0-cos  c».tan2  1(0,  +  02)  +  (i  —  tan2  0)  tan  $(0,  +  02)  = 

tan  0-cos  GO  ; 

/.  tan  i(0,  +  02)  = 

tana0—  i  ±  |/jtan40  —  2  tan8  0  -f  4  tan2  0  -  cosa  o?  -f  1  1 
2  tan  0-cos  G? 

The  upper  sign  is  to  be  taken  for  the  radical  ;  for  suppose  that 
GO  =  o,  then 


which  is  evidently  correct  with  the  upper  sign  only.     Hence, 
as  2  cos2  G?  —  i  =  cos  2&>,  we  have  finally 

tan  i(0,  +  0.) 
tan3  0—  i  +  y  {  tan4  0+2  tana  0  -  cos  2a?  +  I  } 

__  —•  -  -  --  .    • 

2  tan  0-cos  GJ 

as  the  desired  expression  connecting  0,  ,  03  ,  GJ,  and  0. 

To  find  what  value  of  OD  is  negligible,  we  may  substitute 
successively  values  of  GJ  =  i°,  2°,  etc.,  with  0  =  30°  and  60°, 
since  the  error  increases  with  0  above  and  below  45°,  being 
negative  below  45°.  We  may  plot  these  values  and  inter- 
polate or  may  interpolate  directly.  That  value  of  co  would  be 


152 


SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 


negligible  which  would  give  such  a  value  of  tan  J(01  -f-  0a)  for 
0  =  60°  that 

tan  £(0a  +  03)/tan  0  =  I  ±  0.00033. 

This  will  be  found  to  be  GO  =  2°.r.  Thus  if  the  plane  of  the 
coils  is  within  about  2.°  of  the  meridian  the  error  will  be 
negligible  between  0  =  30°  and  0  =  60°.  It  is  not  difficult  to 
adjust  to  this  closeness. 

By  substituting  0  =  45°  in  [120],  less  conveniently  in  [121], 
we  have 


tan 


2  COS  GO 


2  COS  GO 


but  tan  45°  =  I.  Hence  at  0  =  45°  the  mean  angle  J(0,  +  02) 
is  correct  whatever  the  value  of  GO.  That  is,  0a  is  as  much  too 
large  as  02  is  too  small,  or  vice  versa.  This  rather  surprising 
result  is  decidedly  important  as  indicating  that,  so  far  as  this 
source  of  error  is  concerned,  45°  is  the  best  angle  by  far  to  use 
in  accurate  work,  since  it  eliminates  wholly  the  error  due  to 
any  imperfect  adjustment  into  the  meridian.  The  proof  of 

this  proposition  may  be  much 
more  easily  arrived  at  graphically. 
Fig.  3  corresponds  in  all  respects 
with  Fig.  2  except  that  it  is 
drawn  for  0  =  45°,  so  that  F  = 
H.  P  therefore  falls  upon  the 
circumference  of  a  circle  passing 
through  A,  A',  B,  B',  etc.  The 
angle  APA'  =  20  is  measured 
geometrically  by  half  the  semi- 
circumference  ABA'.  When  G?  = 
BOA,  the  angle  BPB'  =  0j  -f  02 ,  and  is  measured  geometri- 
cally by  half  the  arc  BDA'B',  which  is  also  a  semi-circumference. 
Hence  0a  -|-  0a  =  20.  A  similar  statement  is  true  for  any 
other  value  of  G?,  thus  proving  the  proposition. 

As  above  stated  the  most  convenient  way  of  adjusting  the 
coils  to  the  meridian  is  by  sending  the  same  current  first  in  a 
positive  and  then  in  a  negative  direction  through  the  coil,  and 


TANGENT  GALVANOMETER.  153 

adjusting  the  coil  until  the  deflections  are  equal.  This  is  most 
readily  done  as  follows.  Set  the  galvanometer  up  approxi- 
mately. Read  the  index  with  no  current,  calling  this  the  zero 
reading.  Send  a  current  which  deflects  the  needle  by  about 
45°.  Let  the  deflection  corrected  for  zero  be  called  0a.  Re- 
verse the  current  and  let  the  corrected  deflection  be  denoted 
by  02.  Suppose  02  to  be  the  greater,  then 


=  0*  -  0, 


co  being  always  on  the  side  of  the  largest  deflection.  To  ad- 
just, open  the  circuit  and  when  the  needle  is  at  rest  turn  the 
coils  through  an  angle  GO  =  0a  —  (f)l  toward  the  side  of  the 
smallest  deflection.  The  adjustment  will  then  be  very  nearly 
right.  It  is  best  to  take  a  new  zero  reading  and  repeat  as  a 
check  or  for  closer  adjustment.  This  method  does  not  elimi- 
nate the  effect  of  initial  torsion  of  the  fibre. 

The  proof  is  as  follows.     From  the  foregoing  demonstra- 
tion by  substituting  0  =  45°,  tan  0  =  i,  we  have 

cos  GO 

tan  0.  =  -  :  -  =  cos  GO  —  sin  GO,  approx.,  as  GO  is  small, 
I  -|-  sm  G? 

cos  GO 

tan  0a  =  -  -  :  -  =  cos  GO  -4-  sm  GO,  approx.,  as  GO  is  small, 
I  —  sin  GO 

.'.  tan  0a  —  tan  0,  =  2.  sin  GO. 

Let  A  —  0a  —  0.     Then  as  0  =  45°  we  have  by  the  above 
proposition  of  Fig.  3,  0  —  0t  =  A  also. 

/.  0a  =  0  -[-  A,        and         01  =  0  —  A. 
.*.  tan  (0  +  ^)  —  tan  (0  —  A)  =  2,  sin  GO 
Now  as  A  is  small  and  tan  0  =  i 

tan  0  +  tan  A        \-\-A 
tan  (0  +  J)  =  ^tan^.tanj  =  |±3  aPProx-  =  I  +  ^  app. 

tan  0  —  tan  A        i  —  A 
tan  (*  -  J)  =  I+tan<A.tanj  =  ^  approx.  =  ,  -  zA  app. 


154 


SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 


.:  tan  (0  +  A)  —  tan  (0  —  A}  =  4 A  =  2  sin  GO  =  200  approx. 
Now         03  —  0,  =  2^.         .-.  c0  =  02  -  0, ,     Q.E.D. 

/n. — For  inclination  of  mirror  to  horizontal.  In  the  gal- 
vanometer here  considered  the  graduated  circle  is  supposed  to 
be  secured  to  the  surface  of  the  mirror  and  the  mirror  to  be 
placed  horizontally  below  the  needle  with  its  index.  Since 
the  index  always  covers  its  image  when  a  reading  is  taken,  the 
plane  of  sight,  i.e.,  the  plane  containing  the  eye  and  the  index, 
is  always  perpendicular  to  the  mirror.  But  the  index  and 
needle  necessarily  deflect  about  a  vertical  axis.  If,  therefore, 
the  mirror  and  circle  be  not  horizontal  but  inclined,  the  angle 
read  off  upon  it  will  not  be  the  true  angle  <p  swept  through  by 
the  needle,  but  an  oblique  projection  of  that  angle.  It  is,  then, 
essential  to  know  how  nearly  horizontal  the  mirror  must  be  to 
avoid  sensible  error  from  this  cause. 

In  Fig.  4  let  the  circle  FA'J'A"  show  the  graduated  circle 

J 


in  horizontal  projection  and  MN  its  vertical  projection.     Sup- 
pose the  needle  to  be  at  A.     Then  if  index  and  circle  are  both 


TANGENT  GALVANOMETER.  155 

horizontal,  the  circle  would  be  as  shown  in  Fig.  4,  and  the 
plane  swept  through  by  the  index  would  be  shown  by  a  hori- 
zontal line  through  A.  But  if  the  mirror  were  inclined,  then 
the  plane  of  the  index  would  still  be  horizontal,  but  that  of 
the  mirror  would  be  inclined  to  it.  The  result  as  far  as  angular 
readings  are  concerned,  however,  would  be  the  same  as  though 
the  mirror  remained  horizontal  and  the  plane  swept  through 
by  the  index  were  inclined.  It  is  more  convenient  to  represent 
the  latter  in  the  drawing.  Therefore  let  IAJ  represent  the 
vertical  projection  of  the  plane  of  the  index  inclined  to  the 
mirror  at  an  angle  AIL  =  h. 

First.  Suppose  the  index  when  undeflected  to  stand  in  the 
direction  A  A"  .  In  vertical  projection  it  will  appear  as  a 
point  at  A.  Let  it  be  deflected  through  an  angle  whose  true 
value  is  0,  but  which  is  read  on  the  circle  as  0'  =  A'OB'  . 
What  is  the  relation  between  0  and  0'?  At  A'  draw  a  tan- 
gent A'B',  and  prolong  B"OB'  to  intersect  this  at  B'.  Then 

A'B' 


Project  B'  upward  to  B  on  IJ.     Then 

AB, 

tan<^_, 

for  AB  is  the  true  length  of  the  tangent  at  A  cut  off  by  the 
pointer  and  shown  in  horizontal  projection  in  A  B'  ,  and  OA' 
is  shown  in  its  true  length.  Therefore 

tan  0       AB       AB         i 


A'B'  ~BD~  cosk 


.-.  tan  0  =  -  l-7  tan  0'  ; 

\,\J;3  fir 


and  the  correction  factor  for  the  inclination  of  the  mirror  is,  in 
this  case,  i/cos  h,  which  is  the  same  as  though  the  coils 
were  inclined  as  in  the  cosine  galvanometer. 


156  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

Second.  Suppose  that  the  zero  position  of  the  index  were 
/'/'  ,  IJ.  Let  the  needle  be  deflected  through  any  angle  shown 
in  projection  upon  the  mirror  as  0'  =  I  '  OKf.  Thus 

I'K' 
tan  0'  =  TQ. 

As  the  tangent  line  of  which  I'K'  is  the  projection  is  parallel 
to  the  mirror  it  appears  in  this  projection  at  its  full  length  and 
in  the  vertical  projection  as  a  point  at  /.  Then 

I'K1 

tan  0  =  -, 


since  AI  is  the  true  length  of  the  side  of  the  angle  0  which 
appears  as  OF  in  the  mirror  projection.  Therefore 

tan  0       I'O      IL 

"^  =  Tr  =  -AT  -  cos  ^       •     •     • 
tan  0'      AI       AI 

Hence  the  correction  factor  for  the  inclination  of  the  mirror  is 
in  this  case  cos  h. 

The  first  of  these  cases  is  where  the  mirror  is  tipped  about 
a  horizontal  line  through  the  zero  points  ;  the  second,  where 
the  tipping  is  about  a  line  through  the  90°  points.  For  a  tip- 
ping about  any  other  line  the  effect  could  be  ascertained  by 
resolving  it  into  two  parts,  one  with  reference  to  each  of  the 
above  positions.  The  effect  will  be  intermediate  between  the 
two  above  extremes,  so  that  it  need  not  be  further  considered. 

What  value  of  h  will  produce  the  limiting  negligible  error? 
As  h  is  very  small  we  may  write  cos  h  =  I  —  x,  where  x  is  a 
small  fraction.  Then  for  the  first  case 

i  i 

-  7  =  -  —  \-\-x  approx. 
cos  h      i  —  x 

For  the  second  case  we  have  simply 
cos  h  =  i  —  x. 


TANGENT  GALVANOMETER.  1 57 

As  these  enter  as  direct  factors  we  must  have  for  negligibility 

*=  o.oo  033. 

/.  cos  h  =  i  —  x  =  0.99  967 ; 
.-.        A=I°.S. 

This  error  can  be  rendered  negligible  without  difficulty  by 
due  care  in  the  original  construction  of  the  instrument  and  by 
proper  levelling  at  the  time  of  use.  The  most  convenient 
method  for  the  latter  is  to  have  a  plumb-line  hanging  from  a 
marked  point,  and  arranged  to  be  brought  over  a  reference 
point  on  a  plate  attached  to  the  lower  part  of  the  coil,  pains 
being  taken  to  see,  once  for  all,  that  the  mirror  is  level  and  the 
coil  vertical  when  the  line  so  indicates.  It  should  be  noted 
that  neither  reversal  nor  reading  both  ends  of  the  needle  tends 
to  eliminate  this  error. 

Summary. — The  following  table  gives  a  summary  of  the  re- 
sults. By  bringing  the  adjustments  or  by  measuring  the  quan- 


No. 

-*£- 

Are  Required:— 

I 

O.OO2 

8H/H=o.oo2 

Earth's  field. 

2 

0.00  1 

ds/s  =.  o.ooi 

Circumference  of  coil. 

3 

0.000 

dn  =  zero 

Turns  in  coil. 

4 

O.OOI 

50  =  o°.o3 

Deflection. 

5 

Negligible 

(  8(26)  =3.  mm.,  ) 
1  d(2d)  =3.  mm.    \ 

Coil  section. 

6 

8(2!)  =1.3  mm. 

Length  of  needle. 

7 

a  —  4.  mm. 

Centering  of  needle. 

8 

da  =  o°.io 

Torsion. 

9 

0  =  i°.5 

Coil  out  of  vertical. 

10 

GO  =2°.0 

Coil  out  of  meridian. 

ii 

A=i°.5 

Mirror  out  of  horizontal. 

tities  designated  within  the  limits  given  in  the  table,  the  residual 
errors  from  all  the  sources  except  the  first  four  may  be 
made  negligible  with  reference  to  8C/C  =  o.ooi  from  each. 
If  their  residuals  were  all  present  at  this  maximum  amount  cf 
0.0003  eacn  their  resultant  effect  would  be  o.oo  033  V?  —  00086, 


I $8  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

Apart  from  //,  the  only  remaining  sources  of  error  are  2  and  4, 
whose  resultant  effect  would  be  o.ooi  V^=  0.0014,  compared 
with  which  the  above  amount  of  o.oo  086  is  not  quite  negligible 
although  the  actual  amount  probably  would  be.  But  as 
above  given  the  total  value  of  A C/C  would  be  o.oo  033"  X  7 
+  0.001'  X  2  =  0.0017".  Hence  we  ought  to  be  able  with  such 
an  instrument  to  obtain  relative  measurements  with  an  accuracy 
of  about  0.2  per  cent.  And  if  H  can  be  measured  and  relied 
upon  to  0.2  per  cent  as  above  assumed,  we  ought  to  be  able  to 
obtain  absolute  measurements  to  0.2  t/2  =  0.28  or  say  0.3  per 
cent. 

Best  Range  of  Deflections.- — It  is  commonly  stated  that  the 
best  deflection  at  which  to  use  a  tangent  galvanometer  of 
the  kind  above  discussed  is  45°.  This  statement  is  insufficient 
and  not  wholly  exact.  What  we  wish  to  know  is,  what  angle  0 
or  what  range  of  deflections  will  give  the  greatest  fractional  pre- 
cision in  the  current  causing  them,  as  further  explained  at 
page  1 10.  To  determine  this  we  have  to  consider  the  nature  of 
those  errors  which  are  a  function  of  the  deflection,  or  which 
otherwise  affect  6C/C.  These  are  such  as  result  from  1st,  #0; 
2d,  length  of  needle  ;  3d,  torsion  ;  4th,  coil  out  of  meridian.  Of 
these  the  2d  and  3d  can  readily  be  so  corrected  as  not  to  enter 
sensibly,  but  still  the  residual  error  from  the  2d  will  be  least  at 
26°.6,  and  that  from  the  3d  will  diminish  as  the  angle  is  smaller. 
The  error  from  the  4th  source  can  be  eliminated  by  some 
care,  but  is  more  difficult  of  removal  than  the  two  preced- 
ing. Its  residual  is  least  at  45°,  thus  pointing  to  that  angle 
as  the  best.  The  value  of  #0  is  constant  as  far  as  errors  of  eye 
estimation  are  concerned,  and  the  other  sources  of  error  making 
it  up  follow  the  law  of  accidental  distribution.  We  must  there- 
fore treat  #0  as  constant  for  all  values  of  0.  The  following  table 
gives  the  values  of  8C/C  for  #0  —  o°.O3  for  each  5°  from  o°  to 
90°.  Inspection  of  these,  or  better  of  a  plot  made  from  them, 
shows  that  dC/C  is  sensibly  constant  between  0  =  30°  and 
0  =  60°  although  the  minimum  is  at  0  —  45°.  The  precision 
is,  however,  only  about  one-half  less  at  20°  and  70°,  but  beyond 
those  points  it  falls  off  rapidly.  Hence  as  far  as  this  source  of 


ELECTRO-STATIC  CAPACITY. 


159 


error  is  concerned,  any  deflection  between  30°  and  60°  is  equally 
good,  and  between  20°  and  70°  nearly  as  good  ;  but  below  20° 
and  above  70°  should  not  be  used  in  careful  work. 


6  — 

SC      2«0 

6  — 

SC     2&}> 

C   sin  2<f> 

C   sin  2$ 

o°  or  90° 

oc 

25°  or  65° 

±0.00131 

5  or  85 

±0.00589 

30  or  60 

.00116 

10  or  80 

.00292 

35  or  55 

.00107 

15  or  75 

.00200 

40  or  50 

.00102 

20  or  70 

.00156 

45  or  45 

.OOIOO 

Combining  all  the  considerations,  then,  we  see  that  for  the 
very  best  work  0  =  45°  is  preferable ;  that  any  deflection  be- 
tween 30°  and  45°  is,  however,  nearly  as  good  as  45°,  and  that 
any  deflection  between  30°  and  60°  is  but  slightly  inferior  to 
these.  For  most  work  then  it  is  indifferent  as  far  as  6C/C  is 
concerned  what  deflection  we  use  between  the  limits  of  30°  and 
60°.  For  work  somewhat  inferior  in  accuracy  we  may  use  in- 
differently any  angle  between  20°  and  70°,  but  should  rarely  go 
outside  those  limits. 

Example  XXXV,  —  Electro- Static  Capacity.  —  Thomson's 
or  Coifs  Method. — Description  of  method  in  Physical  Labora- 
tory Notes  or  in  Kempe's  Handbook  of  Electrical  Testing. 
The  formula  for  the  method  is 


F  -  —  -F 

— 


[125} 


The  battery  power  used  is  assumed  to  be  sufficient  to  enable 
a  change  equal  to  the  smallest  coil  in  the  resistance-box  to  be 
perceived,  hence  dR  =  SRX.  The  charges  in  the  condensers 
are  greatest  when  R  +  Rx  is  as  large  as  possible,  namely,  the 
total  resistance  r  in  the  box. 
By  formula  [52] 


This  will  be  a  minimum  when  the  last  parenthesis  is  so.     As  R 


l6o  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

and  Rx  are  independent  we  must  differentiate  with  respect  to 
each  successively  and  equate  the  coefficients 


Hence 


The  best  ratio  then  is  R/RX  =  i,  and  therefore  F/FX  =  i  or 
F  •=.  Fx\  that  is,  it  is  best  to  use  a  known  condenser  of  as 
nearly  as  possible  the  value  of  the  unknown. 

Example  XXXVI.—  Magnetometer.  —  In  measuring  M  -£-  H 
by  the  magnetometer,  the  deflecting  magnet  is  placed  succes- 
sively at  two  distances,  r^  and  ra  from  the  needle,  producing 
deflections  0,  and  0a.  And 


M  r/  tan  0,  -  r,»  tan  0, 

-  '         '~ 


Desired,  the  best  ratio  of  ^  :  r,  ;  i.e.,  that  which  will  make 

M 

—  a  minimum.     The  measurements  are  such  that  tfr  and 


<Sya  may  be  considered  negligible,  and  d  tan  0t  =  ^  tan  02  ;  thus 
these  are  the  precision  conditions.  There  are  no  magnitude 
conditions  which  bear  on  this  problem.  Then 


, 


0a) 
/tan 


{M\         -ra6 
\Hl  ~  r.2  -  r.  '* 


To   make  A  —  a  minimum,  the   fraction   in   the   second 
H 

member  must  be  a  minimum  ;  and  we  wish,  therefore,  to  find 


BATTERY  RESISTANCE  AND  E.  M.  F.  l6l 

the  value  of  r,  :  ra  ,  which  will  make  it  so.  Writing,  then,  rt  :  ra 
=  n,  or  rl  =  nr^  ,  and  substituting  gives 

r?*  +  r?         nr?  +  r.'°  6    n10  +  l 

('i'--O'  '"to'-O'"      ''(a'-')1' 

in  which  we  have  to  find  the  value  of  n  to  produce  a  mini- 
mum. 

Where  the  two  variables  are  independent,  as  rl  and  rz  in 
this  case,  the  following  proposition  may  be  often  of  service. 

Let  u  =  f(x,  y)  where  x  and  y  are  independent  variables 
and  /  is  such  a  function  that  it  may  be  separated  so  that 


[127] 


Then  the  value  of  x  :  y,  which  makes  cr  (  —  )  a  minimum,  is 

\yl 

the  same  as  will  make  u  a  minimum.     For  x  cannot  be  ex- 
pressed as  a  function  of  -,  and,  therefore,  p  (x)  does  not  enter 

y 

into  the  determination  of  x  :  y  for  the  minimum.     The  same 
is,  of  course,  true  if  the  separation  be  made  into 


lias] 


Then  in  the  problem  in  hand  we  have  to  find  the  value  of 

n,  which  will  make  £        \*  a  minimum. 
(n  —  i) 

~          =  3^°  -  5«8  -  2  =  o.     .-.  n  =  1.32  approx. 


Example  XXXVII.—  Battery  Resistance  and  E.  M.  F.— 

In  the  ordinary  method  of  measuring  the  resistance  B  of  a 
battery,  the  currents  cl  and  c^  produced  by  the  battery  through 
two  known  external  resistances  rl  and  ra  are  observed.  Let  pl 
and  />„  represent  the  total  resistances,  including  battery,  leads, 


1 62  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

galvanometer,  and   rheostat ;    and   let  E  denote  the  battery 
E.  M.  F.     Then  the  formula  for  the  method  is 


[129] 


Desired  the  best  ratio  of  cl  :  cy 


Following  the  procedure  of   page  105,  the  expression  for 
AB  must  first  be  obtained. 


dB_  _  —  (c,  —  c^r,  —  (c,r,  -  ^,)  _  cfa  -  r,)  _  c,(p,  -  p,) 
dc,  ~  (c,  -  c$  -  fo  -  <:,)'      -  (c,  -  c$  ' 

But  as  c  is  a  function  of  p  and  the  constant  E  the  solution  may 
be  made  either  by  writing  E/c  for  p  or  E/p  for  c.  The  latter 
will  be  done.  Then 


dB  A2p,  o-    M    i        dB  p,p* 

-T-  =  —  TT  ^     Similarly,     —  -  =  _/  ™  a  —  -. 

dc, 


This  expression  will  serve  for  a  galvanometer  for  which  6c  is  a 
constant,  e.g.,  a  reflecting  galvanometer,  or  one  whose  scale  is 
uniform  and  proportional  to  the  current,  such  as  a  Weston 
ammeter.  For  a  tangent  galvanometer  or  any  instrument  for 
which  dc/c  is  a  constant,  the  expression  must  be  modified  to 
change  8*c  to  tfc/c*.  This  may  be  done  by  substituting  in  the 
second  member  E/cl  for  pl  and  E/ct  for  pa.  This  gives 


BATTERY  RESISTANCE  AND  E.  M.  F.  163 

1st.  For  the  reflecting  or  the  Weston  galvanometer  to  find 
the  best  ratio  x  =  cjc^  substitute  in  [130]  the  equivalent  of  x  = 
cjc*  ,  viz.,  x  —  pjpr  This  gives 


_   _   [132] 

Hence  for  a  minimum 


.:  x*  —  2x*  ~  i  =  o, 

for  which  by  approximate  solution  x  =  2.2.     The  best  ratio  is 
then  p3/p1  —  2.2  or  cjc^  —  2.2  or  cl  =  2.2  cv 

To  find  the  best  external  resistances  corresponding  to  this 
best  ratio  we  may  substitute  x  =  2.2  in  [132],  giving 


=  26v-*V,  approx; 


Hence,  as  far  as  this  can  show,  p1  should  be  as  small  as  possi- 
ble. It  must,  however,  be  remembered  that  if  p  is  small  com- 
pared with  B,  a  serious  error  will  enter  from  polarization.  We 
should  therefore  use  as  small  an  external  resistance  as  is  con- 
sistent with  the  polarization  occurring  in  the  given  battery. 

To  give  a  stated  precision  AB/B  the  galvanometer  must 
have  such  sensitiveness  that  it  can  measure  the  current  c,  with 
the  precision 

*£>-  __L-  —  — 

~^<  1/20  "pT^~* 
This  expression  is  of  course  deduced  directly  from  the  above. 


164  SOLUTIONS  OF  ILLUSTRATIVE  PROBLEMS. 

2d.  For  the  tangent  galvanometer  or  any  for  which  dc/c 
is  a  constant,  to  find  the  best  ratio  cjcv  If  we  proceed  by  the 
usual  method  starting  with  [131],  we  shall  arrive  at  the  contra- 
diction i  =  o,  showing  that  there  is  no  such  best  ratio.  But 
we  may  deduce  the  result  which  we  desire  as  follows.  Simplify- 
ing [131],  we  obtain 


,  ~  P 


—         C  \  \      C 


From  the  latter  it  will  be  seen  that  AB  diminishes  as  p,  in- 
creases and  as  pl  diminishes.  Hence  /o2  should  be  as  large  and 
p,  as  small  as  conditions  of  range  of  galvanometer  and  of 
polarization  will  permit.  We  should  therefore  use  by  pref- 
erence deflections  of  60°  and  30°  or  of  70°  and  20°  on  the 
tangent  galvanometer. 

The  necessary  precision  and  sensitiveness  of  galvanometer 
would  be  determined  as  follows.  Using  60°  and  30°,  cl  :  c^  — 
3  :  i  approx.  Then 


$c__i  BAB 
c   <  2  ^  B 

Therefore  we  must  use  a  galvanometer  of  a  precision  at  least 
equal  to  this  value  of  dc/c,  and  of  such  a  "factor"  that  with 
the  smallest  value  of  pJB  admissible  on  account  of  polariza- 
tion, the  deflection  will  be  about  60°.  The  value  of  pa  must, 
of  course,  be  such  as  to  make  the  second  deflection  about  30°. 
If  instead  of  taking  two  deflections  on  one  galvanometer 
we  take  the  second  deflection  on  a  much  more  sensitive  galva- 
nometer than  the  first,  but  equally  precise,  that  is  one  which 


BA  TTER  Y  RESISTANCE  AND  E.  M.  F.  165 

will  measure  a  very  much  smaller  current  but  with  equal 
precision  dc/c,  we  may  then  make  pa  many  times  as  great  as  pl 
and  thus  improve  the  conditions  of  working.  This  is  really 
what  we  do  in  using  a  potential  galvanometer  and  a  current 
galvanometer  in  combination  as  in  the  method  described  in 
the  Physical  Laboratory  Notes. 

E.  M.  F. — By  similar  demonstrations  we  may  show  for 
measuring  the  electromotive  force  of  the  battery  the  following 
points. 

1st.  For  a  galvanometer  for  which  dc  is  a  constant  the 
best  ratio  of  cjc^  is  sensibly  the  same  as  for  measuring  B ; 
also  that,  even  using  this  ratio,  AE  increases  with  p, ,  so  that  p, 
should  be  made  as  small  as  is  consistent  with  polarization,  just 
as  in  measuring  B. 

2d.  For  a  galvanometer  where  dc/c  is  constant  we  must 
make  p2/p,  as  large  as  possible,  making  pl  large  enough  to 
avoid  polarization.  Of  course  it  is  clear  that  a  potential 
galvanometer  is  preferable  to  the  two-deflection  method  for 
measuring  E. 


SINES,    COSINES,    TANGENTS. 


NATURAL,. 

LOGARITHMIC. 

Sine. 

Cos. 

Tan. 

Sine. 

Cos. 

Tan. 

0 

0.0 

o.oooo 

1.  0000 

o.oooo 

CO 

0.0000 

—    CO 

0.5 

0.0087 

1.  0000 

0.0087 

7.9408 

o.oooo 

7.9409 

1. 

0.0175 

0.9998 

0.0175 

8.2419 

9.9999 

8,2419 

1.5 

0.0262 

0.9997 

0.0262 

8.4179 

9.9999 

8.4181 

2. 

0.0349 

0.9994 

0.0349 

8.5428 

9.9997 

8.5431 

2.5 

0.0436 

0.9990 

0.0437 

8.6397 

9.9996 

8.6401 

3. 

0.0523 

0.9986 

0.0524 

8.7188 

9.9994 

8.7194 

4. 

0.0698 

0.9976 

0.0699 

8.8436 

9.9989 

8.8446 

5. 

0.0872 

0.9962 

0.0875 

8.9403 

9.9983 

8.9420 

10. 

0.1736 

0.9848 

0.1763 

9.2397 

9-9934 

9.2463 

20. 

0.3420 

0.9397 

0.3640 

9-5341 

9.9730 

9.5611  . 

30. 

0.5000 

0.8660 

0-5774 

9.6990 

9-9375 

9.7614 

40. 

0.6428 

0.7660 

0.8391 

9.8081 

9-8843 

9.9238 

45. 

0.7071 

0.7071 

I.OOOO 

9-8495 

9.8495 

o.oooo 

50. 

o.  7660 

0.6428 

1.1918 

9.8843 

9.8081 

0.0762 

60. 

0.8660 

0.5000 

1.7321 

9-9375 

9.6990 

0.2386 

70. 

0.9397 

0.3420 

2-7475 

9.9730 

9-5341 

0.4389 

80. 

0.9848 

0.1736 

5-6713 

9-9934 

9.2397 

0.7537 

90. 

1.  6000 

o.oooo 

CO 

o.oooo 

—    CO 

CO 

CONSTANTS. 

I  metre  in  inches  (U.  S.  C.  S.,  1892) 

i  inch  in  millimetres 

i  kilogramme  in  pounds  avoirdupois  (U.  S.  legal), 
i  pound  avoirdupois  in  kilogrammes  (U.  S.  legal). 

e  =  base  of  Naperian  logarithms 

logic  €  =  modulus  of  common  logarithms 

Radius  is  equal  in  length  to  an  arc  of 

Arc  of  i°  in  terms  of  radius 

Watts  per  horse-power  (see  p.  132) , 

Small  calories  per  second  per  watt  (see  p.  in). . . . 


=     39-3700 
=    25.40  oo 
=      2.20460 
=      0.45  359  7 
=      2.71  828  18 
=      0.43  429  45 
=  57°-29  578 
=       o.oi  745  329. 
=   746. 
=       0.23  87 


166 


SQUARES,    CUBES,    RECIPROCALS. 


No. 

Square. 

Cube. 

Secip. 

No. 

Square. 

Cube. 

Recip. 

1.0 

.00 

1.  00 

1.  00 

5.5 

30.3 

166. 

.182 

1.1 

.21 

1.33 

0.909 

5.6 

31-4 

176. 

.179 

.2 

.44 

1-73 

.833 

5.7 

32-5 

185. 

.175 

.3 

.69 

2.20 

.769 

5.8 

33-6 

195. 

.172 

.4 

.96 

2.74 

.714 

5.9 

34-8 

205. 

.169 

.5 

2.25 

3.38 

.667 

6.0 

36.0 

216. 

.167 

.6 

2.56 

4.10 

.625 

6.1 

37-2 

227. 

.164 

.7 

2.89 

4.91 

.588 

6.2 

38.4 

238. 

.161 

1.8 

3-24 

5.83 

.556 

6.3 

39-7 

250. 

•  159 

1.9 

3-61 

6.86 

.526 

6.4 

41.0 

262. 

.156 

2.0 

4.00 

8.00 

.500 

6.5 

42.3 

275- 

.154 

2.1 

4.41 

9.26 

.476 

6.6 

43-6 

287. 

.152 

2.2 

4.84 

10.6 

•455 

6.7 

44-9 

301. 

.149 

2.3 

5-29 

12.2 

•435 

6.8 

46.2 

314. 

.147 

2.4 

5.76 

13.8 

.417 

6.9 

47.6 

329- 

.145 

2.5 

6.25 

15-6 

.400 

7.0 

49.0 

343- 

.143 

2.6 

6.76 

17.6 

.385 

7.1 

50.4 

358. 

.141 

2.7 

7.29 

19.7 

•  370 

7.2 

51.8 

373- 

.139 

2.8 

7.84 

22.0 

•  357 

7.3 

53-3 

389. 

.137 

2.9 

8.41 

24.4 

•345 

7.4 

54.8 

405. 

.135 

3.0 

9.00 

27.0 

•333 

7.5 

56.3 

422. 

.133 

3.1 

9.61 

29.8 

.323 

7.6 

57-8 

439- 

.132 

3.2 

IO.2 

32.8 

.313 

7.7 

59-3 

457- 

.130 

3.3 

lO.g 

35-9 

.303 

7.8 

60.8 

475- 

.128 

3.4 

n.6 

39-3 

.294 

7.9 

62.4 

493- 

.127 

3.5 

12.3 

42.9 

.286 

8.0 

64.0 

512. 

.125 

3.6 

13-0 

46.7 

.278 

8.1 

65.6 

53L 

.123 

3.7 

13.7 

50.7 

.270 

8.2 

67.2 

551. 

.122 

3.8 

14.4 

54-9 

.263 

8.3 

68.9 

572. 

.120 

3.9 

15.2 

59-3 

.256 

8.4 

70.6 

593- 

.119 

.0 

16.0 

64.0 

.250 

8.5 

72.3 

614. 

.118 

.1 

16.8 

68.9 

.244 

8.6 

74.0 

636. 

.116 

.2 

17.6 

74.1 

.238 

8.7 

75-7 

659- 

•US 

.3 

18.5 

79-5 

.233 

8.8 

77-4 

681. 

.114 

.4 

19.4 

85.2 

.227 

8.9 

79.2 

705. 

.112 

.5 

20.3 

91.1 

.222 

9.0 

81.0 

729. 

.III 

.6 

21.2 

97-3 

.217 

9.1 

82.8 

754- 

.no 

4.7 

22.1 

104. 

.213 

9.2 

84.6 

779- 

.109 

4.8 

23.0 

in. 

.208 

9.3 

86.5 

804. 

.108 

4.9 

24.O 

118. 

.204 

9.4 

88.4 

831. 

.106 

5.O 

25.0 

125. 

.200 

9.5 

90.3 

857. 

.105 

5.1 

26.0 

133. 

.196 

9.6 

92.2 

885. 

.IO4 

5.2 

27.O 

141. 

.192 

9.7 

94.1 

913. 

.103 

5.3 

28.1 

149. 

.189 

9.8 

96.0 

941. 

.102 

5.4 

29.2 

157- 

.I85 

9.9 

98.0 

970. 

.101 

167 


LOGARITHMS. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.0 

oooo 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

1.1 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

1.2 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

1.3 

"39 

H73 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

1.4 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

1.5 

1761 

1790 

1818 

1847 

1875 

1903 

i93i 

1959 

1987 

2014 

1.6 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

1.7 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

1.8 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

1.9 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

2.0 

3010 

3032 

3054 

3075 

3096 

3H8 

3139 

3160 

3181 

3201 

2.1 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

2.2 

3424 

3444 

3464 

3483 

3502 

3522 

354i 

356o 

3579 

3598 

2.3 

3617 

3636 

3655 

3674 

3692 

37" 

3729 

3747 

3766 

3784 

2.4 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

2.5 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

2.6 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

2.7 

43M 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

2.8 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

2.9 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

3.0 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

3.1 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

3.2 

5052 

5065 

5079 

5092 

5105 

5H9 

5132 

5145 

5159 

5172 

3.3 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

3.4 

5315 

5328 

5340 

5353 

53^6 

5378 

5391 

5403 

54i6 

5428 

3.5 

5441 

5453 

5465 

5478 

5490 

5502 

5515 

5527 

5539 

5551 

3.6 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

3.7 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

3.8 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

3.9 

59" 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

4.O 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

4.1 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

4.2 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

4.3 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

4.4 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

4.5 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

4.6 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

4.7 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

4.8 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

4.9 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

5.0 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

5.1 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

5.2 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

5.3 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

5.4 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

738o 

7388 

7396 

168 


LOGARITHMS. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

5.5 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

5.6 

7482 

749° 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

5.7 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

5.8 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

5.9 

7709 

7716 

7723 

773i 

7738 

7745 

7752 

7760 

7767 

7774 

6.0 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

6.1 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

6.2 

7924 

793i 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

6.3 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

6.4 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

6.5 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

6.6 

8i95 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

»254 

6.7 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

6.8 

8325 

8331 

8338 

8344 

835i 

8357 

8363 

8370 

83/6 

8382 

6.9 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

7.0 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

7.1 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

»555 

8561 

8567 

7.2 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

7.3 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

7.4 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

7.5 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

7.6 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

7.7 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

7.8 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

7.9 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

901  5 

9020 

9025 

8.0 

9031 

9036 

9042 

9047 

9°53 

9058 

9063 

9069 

9074 

9079 

8.1 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9J33 

8.2 

9r38 

9M3 

9149 

9!54 

9J59 

9165 

9170 

9J75 

9180 

9186 

8.3 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

8.4 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

8.5 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

8.6 

9345 

9350 

5355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

8.7 

9395  ' 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

8.8 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

8.9 

9494 

9499 

9504 

9509 

9513 

95i8 

9523 

9528 

9533 

9538 

9.0 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

958i 

9586 

9.1 

959° 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

9.2 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

9.3 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

9.4 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9764 

9768 

9773 

9.5 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

9.6 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

9.7 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

9.8 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

9.9 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

169 


INDEX. 


PAGE 

Accuracy 13,  25,  45 

of  method.     See  "  Error  of  method  " 46 

of  result 13,  35,  45,  84 

Estimation  of 13,  32,  35,  45,  86 

Importance  of  estimate  of i,  36 

Forms  of  Problems  on 33 

A.  D.     See  average  deviation 

Application  of  general  formulae  to  precision  discussions 54 

Average .  16 

Average  deviation 16 

Advantage  of,  over  other  deviation  measures 24 

Examples  of.       Example  II 18 

of  single  observation 16 

of  mean  result 18 

Significance  of 19 

Balance,  Weighing  by  an  equal  arm.     Example  III 37 

Battery  resistance  and  E.  M.  F.     Example  XXXVII 161 

Beam,  Modulus  of  elasticity  of.     Examples  XXVIII,  XXX 115,  118 

Best  distribution  of  labor I,  33,  36,   70 

Best  magnitudes  of  components 47,   100 

Best  ratio  of  components.     See  "  Best  magnitudes  " 47,  100 

Best  representative  value 14,     16 

Best  value  of  n  for  a  series  of  observations 20 

Best  value  of  precision  measure  of  components 47 

Best  value  of  residuals 27 

Calibration  of  voltmeter.     Example  XXXI «. 120 

Calorimeter.     Examples  XVII,  XXII 88,  94 

Capacity,  Electro-static.    Thomson's  and  Gott's  methods.    Example  XXXV.   159 

Check  methods  and  results 8,  47 

Clark  cell,  Calibration  of  voltmeter  by.     Example  XXXI 120 

Collective  effects 50 

Combined  effects 50 

171 


INDEX. 


Components, 

Best  ratio  of.     See  "  Best  magnitudes  "  ..................  47,  100 

Criteria  for  negligibility  of,  or  of  d  in  ......................  67 

Precision  measure  of,  as  related  to  that  of  result  ............  47 

To  find  best  magnitudes  of, 

Single  component  ................  102 

Two  variable  components  ........  104 

Several  components  .........  107,  108 

To  find  best  value  of  precision  measure  of  ..................  47 

Constant  error  ......................................................  7 

Constants, 

Rejection  of  places  of  figures  in  ..........................  .  .,  .  70 

Table  of  ..................................................  166 

746  watts  =  i  horse-power  ..................................  132 

Corrections  .........................................  .................  £-12 

Cosine  galvanometer.     Example  XXI  ..................................  91 

Cosines,  Table  of  ............................  ...........    ...........  116 

Cradle  dynamometer, 

Example  XXVI  ..................................  96 

Efficiency  of  dynamo  by.     Example  XXXIII  ........  130 

Criteria 

for  negligibility  of  components  ................................  67 

S  in  components  ............................  67 

residuals  .......................   ..........  26 

rejection  of  doubtful  observations  ...........................  30 

Cubes,  Table  of  ......................................................  167 

Data  required  to  substantiate  results  .................................  36,  85 

Determinate  errors  ...................................................  10 

Deviations  ...........................................................  14 

Frequency  of  ...............................................  20 

General  law  of  ............................................  15 

Special  law  of  ........................................  ...    .  .  24 

Deviation  measure  ..............................................  14,  16,  23 

Fractional  ......................  .  ................  29 

Negligible  amounts  in  ...........    ...................  40 

of  mean  result  ......................................  18 

of  single  observations  ...............................  17 

Significance  of  .....................................  19 

Significant  figures  in  ................................  17 

Direct  measurements  ........  .........................................  4 

Planning  of  ......................................  36 

Discordance  of  observations  ...................................    ......  6 

Doubtful  observations  ................................................  30 

Criterion  for  rejection  of  .........................  30 


INDEX.  173 

PAGE 

Dynamo,  Efficiency  of  ...............................................   122 

by  cradle  dynamometer.     Example  XXXIII  .......    130 

by  stray-power  method.     Example  XXXII  ........   122 

Efficiency  of  dynamo  .................................................  122 

by  cradle  dynamometer.     Example  XXXIII  ........  130 

by  stray-power  method.      Example  XXXII  .........  122 

E.  M.  F.  and  resistance  of  battery.     Example  XXXVII  ................  161 

Electro-static  capacity.    Thomson's  and  Gott's  methods.    Example  XXXV.  159 

Elimination  of  constant  error  .....................  .....................  7 

Equal  effects, 

Application  to  best  magnitudes  of  components  .............  108 

Demonstration  ..........................................  70 

General  formulae  ........................................  53 

Special  formulae,  following  general  formulae.     See  also/(  )  in 
this  index. 

Error 

of  method  ,  .............................................  ...  .....  46 

of  result  ..................................................  13,  35.  45 

of  single  observation  .......  .  ....................................  6 

Errors, 

Constant  .....................................................  7 

Constant  part  of  ...............................................  7 

Determinate  ................................................  10 

Indeterminate  ...............   .................................  10 

Variable  part  of  .............................................  6 

Estimated  precision  measure  of  component  ...........  ..................  72 

Estimation  of  accuracy  or  error  of  result  ..........................  13,  32,  45 

direct  measurement  ......................................  13 

indirect  measurement  ....................................  45 

Examples.     See  table  of  contents. 

Factors,  separation  of  functions  into  .......................  58,  60,  61,  64,  76 

Forms  of  problems  or  accuracy  of  result  .............................  33,  84 

Formulae  for  general  and  special  functions.     See/()  in  this  index  ........  55 

Frequency  of  deviations  ...............................................  20 

Friction  brake.     Example  XXIII  ........  ............................    ..  96 

/<)• 

—  /(/«!  ,  m*,   .  .  .  ,  mn)  ............................................  55 

=  ±ml±mt±...  ±  mn  ..................  .  ......................  5^ 

=  am,  +  bmi-\-  .  .  .  +  kmn  ........................  ..............  >•  57 

=  a-mm*'  .  .  .  -mn  .............................................  58 


5° 


a>mv  .......  .  ..................................................     59 


1/4  INDEX. 


=  a-m"0'  .  .  .  •  m*w  ...............................................  60 

=  <£(*«,)•  ,X**a)-  ....  <7(m«)  ........................................  61 

=  <£(*«,,  .  .  .  ,  mp)  ±  p(mq,  .        ,  ms)  ±  etc  .........................  63 

...,»/)*  P(mq  ,  .  .  .  ,  m*)  *  etc  ..........................  64 


g.     Examples  of  measurement  of.     XIII-XVI,  XX  ----  .  ..............  86,  90 

General  formulae  for  relation  between  precision  measure  of  result  and  of 

components  ...................................................  48 

General  law  of  deviations  .............................................  15 

Heat  in  conductor.     Example  XXVI  .................................  ill 

incandescent  lamp.     Example  XVIII  ..........................  89 

Horse-power  =  746  watts,  deduction  of  .................................  132 

Indeterminate  errors  .................................................  10 

Indirect  measurements  ........................................  4,  45,  85 

Estimate  of  accuracy  of  .......................  .  .  45 

Planning  of  ......................................  85 

Labor,  Best  distribution  of  ...................................   I,  33,  36,  70 

Laws  of  deviation  ................................................   25,  73 

Magnetometer.     Example  XXXVI  .....................................  160 

Magnitude,  Best,  for  components  ..................................  47,  100 

Mean,  Arithmetical  ....................  .  ...........................    16,  32 

Method,  Error  of  ....................................................  46 

Mistakes,  Criterion  for  rejection  of  ...........    ........................  30 

Modulus  of  elasticity  of  beam.     Examples  XXVIII,  XXX  ..........   115,  118 

^Moment  of  inertia,  Design  of  bar  for.     Example  XXVII  .................  112 

Negligible  amounts:  Negligibility,  Criteria  for, 

in  components  of   indirect 

measurement  ........  67 

in  constants  .............  70 

in  deviation  measures  .....  20 

in  residuals  ..............  26 

Notation  used  in  formulae  .............................................  49 

Numerical  constants  .............................................  70,  166 

Rejection  of  places  in  ............................  70 

Omission  of  terms  in  differentiating  ...................................  75 

Pendulum.     Examples  XIII-XVI,  XX  ...............................  86,  90 

Percentage  accuracy  .................................................  13 

deviation  .................................................  *3 


INDEX.  175 


PAGE 


Percentage  precision 2g 

Places  of  figures, 

Meaning  of 76 

Rules  for 78 

Planning  of  direct  measurement. 36 

indirect  measurement 85 

Precision, 

Definition  of 25 

Fractional  and  percentage 29 

Measure  of 2^,  47 

Precision  discussion,  Application  of  general  formulae  to 54 

Preparation  of  functions  for 74 

Precision  measure  of  components,  Estimated 72 

of  direct  observation . .  25 

of  result  of  indirect  measurement 47 

Application  of 25~33 

Relation    of,    to   pre- 
cision measures  of 

components 47 

Preparation  of  functions  for  precision  discussion 74 

Probable  error 23 

Problems.     See  "  Examples"  in  table  of  contents. 

Publication  of  results,  Data  which  should  be  stated , 36,  85 

Quantities,  Conditioned,  Independent 5 

Reciprocals,  Table  of , 167 

Rejection  of  doubtful  observations 30 

Relation  between  precision  measure  of  results  and  components  47 

General  formulae  for 48 

Special  formulae  for.     See/"( )  in  this  index. 

Types  of  problems 47 

Residuals n 

Best  values  of 27,  33 

Criteria  for  negligibility  of 26 

Equal  effects " 27 

Resultant  effects,  General  formulae  for 50 

Results  of  indirect  measurements,  Relation  between  precision  measure  of 

results  and  of  components 47 

Rules  for  significant  figures 78 

Separate  effects,  General  formulae  for 49 

Separation  into  factors 58,  60,  61,  64,  76 

"     groups 63,  76 

Significant  figures 75 


1/6  INDEX. 

PAGE 

Significant  figures,  Rules  for .. .  78 

Simplification  of  functions 75 

Sines,  Table  of 106 

Sources  of  error 5 

Direct  measurements 5 

Error  of  method 46 

Special  law  of  deviations 25,   73 

Specific  resistance.     Examples  XXIV,  XXIX 98,  118 

Sphere,  Volume  of.     Example  XIX go 

Squares,  Table  of 167 

Steel  tape.     Example  I , . .  9 

Stray-power  method  for  efficiency  of  dynamos.     Example  XXXII 122 

Tables, 

Constants  166 

Logarithms 168,  169 

Sines,  cosines,  tangents 166 

Squares,  cubes,  reciprocals . .    167 

Tangent  galvanometer, 

Best  deflection.     Examples  XXV,  XXXIV. ..   no,   158 

General  discussion.     Example  XXXIV 138 

Tangents,  Table  of i£6 

To  estimate  the  accuracy  of  a  completed  result 35,   86 

To  find  the  precision  measure  of  an  indirect  result  from  those  of  its  com- 
ponents       47 

To  find  the  best  ratio  or  magnitude  of  the  components 47 

value  of  the  precision  measures  of  the  components 47 

To  obtain  a  result  of  specified  accuracy 34,   85 

the  most  accurate  result  practicable 33,  84 

Variable  error 6 

Variable  parts  of  error 6 

Voltage  measurement  by  Weston  voltmeter.     Example  IV 41 

Voltmeter.     Examples  IV,  XXXI 4,  120 

Volume  of  sphere.     Example  XIX 90 

Watts  per  horse-power  =  746,  Deduction  of 132 

Weighing  by  equal-arm  balance.      Example  III 37 

Weighted  mean 31; 

Weights  of  observations 31 

Weston  voltmeter,  Calibration  of,  by  Clark  cell.     Example  IV. .  * 41 


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